A&A 625, A139 (2019) Astronomy https://doi.org/10.1051/0004-6361/201935129 & © ESO 2019 Astrophysics

Thermal properties of slowly rotating asteroids: results from a targeted survey? A. Marciniak1, V. Alí-Lagoa2, T. G. Müller2, R. Szakáts3, L. Molnár3,4, A. Pál3,5, E. Podlewska-Gaca1,6, N. Parley7, P. Antonini8, E. Barbotin9, R. Behrend10, L. Bernasconi11, M. Butkiewicz-B˛ak1, R. Crippa12, R. Duffard13, R. Ditteon14, M. Feuerbach10, S. Fauvaud15, J. Garlitz16, S. Geier17,18, R. Goncalves19, J. Grice20, I. Grzeskowiak´ 1, R. Hirsch1, J. Horbowicz1, K. Kaminski´ 1, M. K. Kaminska´ 1, D.-H. Kim21,22, M.-J. Kim22, I. Konstanciak1, V. Kudak23,24, P. Kulczak1, J. L. Maestre25, F. Manzini12, S. Marks10, F. Monteiro26, W. Ogłoza27, D. Oszkiewicz1, F. Pilcher28, V. Perig24, T. Polakis29, M. Polinska´ 1, R. Roy30, J. J. Sanabria17, T. Santana-Ros1, B. Skiff31, J. Skrzypek1, K. Sobkowiak1, E. Sonbas32, O. Thizy10, P. Trela1, S. Urakawa33, M. Zejmo˙ 34, and K. Zukowski˙ 1 (Affiliations can be found after the references)

Received 25 January 2019 / Accepted 9 April 2019

ABSTRACT

Context. Earlier work suggests that slowly rotating asteroids should have higher thermal inertias than faster rotators because the heat wave penetrates deeper into the subsurface. However, thermal inertias have been determined mainly for fast rotators due to selection effects in the available photometry used to obtain shape models required for thermophysical modelling (TPM). Aims. Our aims are to mitigate these selection effects by producing shape models of slow rotators, to scale them and compute their thermal inertia with TPM, and to verify whether thermal inertia increases with the rotation period. Methods. To decrease the bias against slow rotators, we conducted a photometric observing campaign of main-belt asteroids with periods longer than 12 h, from multiple stations worldwide, adding in some cases data from WISE and Kepler space telescopes. For spin and shape reconstruction we used the lightcurve inversion method, and to derive thermal inertias we applied a thermophysical model to fit available infrared data from IRAS, AKARI, and WISE. Results. We present new models of 11 slow rotators that provide a good fit to the thermal data. In two cases, the TPM analysis showed a clear preference for one of the two possible mirror solutions. We derived the diameters and albedos of our targets in addition to their +33 +60 −2 −1/2 −1 thermal inertias, which ranged between 3−3 and 45−30 J m s K . Conclusions. Together with our previous work, we have analysed 16 slow rotators from our dense survey with sizes between 30 and 150 km. The current sample thermal inertias vary widely, which does not confirm the earlier suggestion that slower rotators have higher thermal inertias. Key words. minor planets, asteroids: general – techniques: photometric – radiation mechanisms: thermal

1. Introduction are covered with thick layer of insulating regolith, resulting in small thermal inertia values. Also, bodies larger than 100 km Thermal infrared flux from asteroids carries information on their in diameter are usually covered with very fine-grained regolith surface regolith properties like thermal inertia, surface rough- (thus displaying smaller thermal inertia), while smaller sized ness, regolith grain size, and their compactness. Thermal data ones display signatures of coarser grains in the surface, resulting also allow us to precisely determine the size and albedo of these in larger thermal inertia (Gundlach & Blum 2013). objects, when coupled with visual absolute magnitudes. Asteroid These properties can provide valuable insights into the sizes are otherwise hard to determine, unless multi-chord stellar dynamical, collisional and thermal history of asteroids, some of occultations, adaptive optics images, or radar echoes are avail- which are intact leftover planetesimals that created planets and ˇ able (Durech et al. 2015), yet they are essential characteristics shaped the solar system. It might also be possible to study deeper for studies of for example the collisional evolution based on cur- layers under the immediate surface by using longer wavelengths rently observed size-frequency distribution (Bottke et al. 2005), that penetrate deeper, or by studying targets with long episodes asteroid densities (Carry 2012), or asteroid family members dis- of the Sun heating the same surface area, like slow rotators or persion under thermal recoil force (Vokrouhlický et al. 2015). targets with low spin axis inclination to the orbit. Physical properties of the regolith covering asteroid surfaces are For detailed studies of asteroid thermal properties, various connected to their age and composition. Young, fresh surfaces thermophysical models (henceforth: TPM) are being used (for display high thermal inertia, because of the small amount of fine- their overview see Delbo’ et al. 2015). The essential input to grained regolith on the surface, while the surfaces of old targets apply these models is the knowledge of the spin and shape of ? Full Table A.1 and photometric data from all individual the objects under study. The rotation period and spin axis incli- nights are only available at the CDS via anonymous ftp to nation dictate the duration and intensity of the alternating cycles cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc. of surface heating and cooling. Such cycles, when especially u-strasbg.fr/viz-bin/qcat?J/A+A/625/A139 intense, can result in thermal cracking (Delbo’ et al. 2014). As

Article published by EDP Sciences A139, page 1 of 40 A&A 625, A139 (2019) for the shape, at first approximation it can be represented by periods with growing heliocentric distance (Marzari et al. 2011). a sphere, although such a simple shape often fails to explain Also, as has been stressed by Warner & Harris(2011), as much thermal lightcurves or separate thermal measurements taken at as 40% of lightcurves considered in the Lightcurve Database 1 different viewing aspects and phase angles. (LCDB) as reliable, have small maximum amplitude: amax ≤ Today the availability of spin and shape modelled targets is 0.2 mag, posing additional challenges in period determination the main limiting factor for asteroid studies by TPM. Also, in due to potential ambiguities, and having profound effects on spite of an abundance of thermal data available mainly from the results from any survey using asparse-in-time observing WISE (Wide-field Infrared Survey Explorer), AKARI, IRAS cadence. (the Infrared Astronomical Satellite), or Herschel space obser- Some light on the subject of biases in asteroid models was vatories, they also possess their limitations: for unique solutions shed by recent results based on data for asteroids from Gaia from thermophysical modelling, thermal data have to come from Data Release 2 (DR2), which facilitated unique determination of a range of aspect and phase angles, probing the target rotation spin parameters for around 200 targets (Durechˇ & Hanuš 2018). with sufficient resolution at the same time. As a result, so far This first, largely unbiased sample of targets (with some excep- detailed TPM have been applied to less than 200 asteroids (see tions, see Santana-Ros et al. 2015) modelled using very precise the compilations by Delbo’ et al. 2015; Hanuš et al. 2018a), absolute brightnesses revealed interesting results. Half of the and only a small fraction of them rotate slowly, with periods asteroids in this sample turned out to rotate slowly, with periods exceeding 12 h. Slow rotators are especially interesting in this longer than 12 h, unlike in the sets of models from the majority context, as a trend of increasing thermal inertia with period has of previous surveys, based on ground-based data, where shorter been found by Harris & Drube(2016), based on estimated beam- periods dominated (e.g. Hanuš et al. 2011, 2013). Results from ing parameters (η) determined on data from the WISE space Gaia DR2 confirm the findings of Szabó et al.(2016) and Molnár telescope (Wright et al. 2010). et al.(2018) based on data from the Kepler Space Telescope The distribution of thermal inertia amongst the members of of the substantial contribution of slow rotators in the asteroid asteroid families would be crucial for example in the search for population. evidence for asteroid differentiation, separating iron rich from It should also be noted that models based on sparse data iron poor family members (Matter et al. 2013). This quantity provide reliable spin parameters, but only low-resolution, coarse is also essential for estimates of the orbital drift caused by the shape representations (flag “1” or “2” in the shape quality sys- Yarkovsky effect, and for studies of regolith properties. How- tem proposed by Hanuš et al. 2018b), limiting their use in other ever, the relatively small number of targets with known spin and applications (Hanuš et al. 2016). For example, detailed ther- shape parameters prevents detailed thermophysical studies on mophysical modelling needs rather high-resolution (flag “3”) large groups of asteroids, limiting these to using simple thermal shape models as input, only possible with large datasets of dense models with spherical shape approximation, often insufficient to lightcurves obtained at various viewing geometries. Such mod- explain thermal data taken in various bands and viewing geome- els usually provide a much better fit to thermal data than shapes tries. It is especially vital in the light of high precision thermal approximated by a sphere with the same spin axis and period infrared data from the WISE and Spitzer space telescopes now (Marciniak et al. 2018). available, with uncertainties comparable to average uncertainties In the next section we refer to our targeted survey, then of the shape models based on lightcurve inversion (Delbo’ et al. in Sect.3 we describe our modelling methods. Results from 2015; Hanuš et al. 2015). both lightcurve inversion and thermophysical modelling are pre- Nowadays asteroid spin and shape models are created mainly sented in Sect.4, and discussion and conclusions in Sect.5. from sparse-in-time visual data (of the order of one to two points AppendixA contains details of the observing runs and com- per day) from large surveys like those compiled in the Lowell posite lightcurves, while AppendixB presents O–C plots from photometric database (Hanuš et al. 2011). The low photometric thermophysical analysis. precision of such data (average σ ∼ 0.15–0.2 mag), originally gathered for astrometric purposes, results in the favouring of some targets in the modelling, for example mainly those dis- 2. Targeted survey playing large amplitudes in each apparition (Durechˇ et al. 2016). Taking all the above-mentioned facts into account, it is clear A partial solution to the problem is joining sparse data with that there is a continuing need for targeted, dense-in-time obser- dense lightcurves (like in e.g. Hanuš et al. 2013), an approach vations of objects omitted by previous and ongoing surveys. limited though by the availability of the latter, which are also Such targets are mainly slow rotators, and also bodies that strongly biased towards large amplitudes and short periods. On either constantly or temporarily display lightcurves of small the other hand, targeted surveys gathering dense lightcurves amplitudes. To counteract both selection effects, our observ- favour certain types of objects, like near Earth asteroids (NEAs) ing campaign is targeted at main-belt slow rotators (P > 12 h), or members of specific groups or families like Flora, Eos, or which at the same time have small maximum amplitudes Hungaria asteroids. The resulting spin and shape determinations, (amax ≤ 0.25 mag). Neither sparse data from large surveys nor even if taken altogether, do not provide unbiased information available datasets of dense lightcurves allow for their precise on for example the spin axis distributions of the whole aster- spin and shape modelling, so they require a dedicated observ- oid population. Instead, they are biased by spin clusters within ing campaign. It is worth noting that it is exactly these features – families (Slivan 2002; Kryszczynska´ 2013) and by preferential small and slow flux variations – that make them challenging for retrograde rotation of NEAs (Vokrouhlický et al. 2015). Taking spin and shape reconstructions, which make these targets per- all this together can distort the results for the whole popu- fect calibration sources for a range of infrared observatories (like lation, missing certain specific cases, so that many elements ALMA – Atacama Large Millimeter Array, APEX – Atacama essential for properly understanding asteroid dynamics, physics, Pathfinder Experiment, or IRAM – Institut de Radioastronomie and evolution are lacking (Warner & Harris 2011). For exam- Millimétrique, Müller & Lagerros 2002), on the condition that ple, Jupiter Trojans on average rotate much more slowly than main belt asteroids, so there might be a gradation of rotation 1 http://www.MinorPlanet.info/lightcurvedatabase.html

A139, page 2 of 40 A. Marciniak et al.: Thermal properties of slowly rotating asteroids these variations can be exactly predicted. To this purpose their 3. Lightcurve inversion and thermophysical reliable spin and shape model are necessary. modelling The details of our campaign conducted at over 20 stations from 12 countries worldwide, the target selection procedure, and For spin and shape reconstructions we use lightcurve inversion first results are described in Marciniak et al.(2015, 2016). In by Kaasalainen & Torppa(2001), which represents the shape a nutshell, our survey revealed a substantial number of slow model by a convex polyhedron. All the lightcurves were treated rotators which, in spite of having good data from dense observa- as relative, and not absolutely calibrated, as from our campaign tions, had wrongly determined rotation periods, summarised in we obtain mostly relative brightness measurements. Only the LCDB (Warner et al. 2009), a database used in a variety of fur- models with a unique solution, clearly the best in terms of χ2 fit ther studies on asteroid spins and dynamics. We corrected those of observed to modelled lightcurves, have been accepted. Each of determinations with the new values for the periods, and con- the acceptable shape model solutions has been visually inspected firmed them in consecutive apparitions. We also constructed spin to fulfil the criterion of rotation around the axis of greatest iner- and shape models for the first sample our targets, using dense tia. From the range of accepted solutions the uncertainty range lightcurves from multiple apparitions, and scaled these mod- of spin axis position and period has been evaluated. As shown els by stellar occultations fitting and thermophysical modelling, by Kaasalainen et al.(2001) this is the only practical way of with consistent size determinations (Marciniak et al. 2018) In the obtaining realistic uncertainty for spin axis position because for- latter work we also presented simultaneous fits to data from three mal errors like those obtained from the covariance matrix tend different infrared missions (IRAS, AKARI, and WISE), and in to be strongly underestimated. The effects of random noise are spite of potential calibration problems of each of the missions much smaller than systematic errors not uncommon in photo- alone and cross-calibrations between them, we obtained good metric data, or the combined effects of model shape and spin fits. This gave us confidence in our models and methods used. axis uncertainties. The uncertainty of period is also determined The most important of our previous results presented in from the range of best-fit solutions, and is dictated by the time Marciniak et al.(2018) were large thermal inertia values obtained span of the whole photometric dataset and the period duration for most of our slow rotators, which followed the trend found by itself. If the minimum in the periodogram is substantially lower Harris & Drube(2016). This seemed to support the idea that for than the others, the period uncertainty can be assumed to be one slow rotators we observe thermal emission from deeper, more hundreth of this minimum width, which is usually of the same compact layers of the surface, a fact that might open new paths order as the one determined from the best-fit solutions. for studies of asteroid regolith. However, due to the small num- As is usually the case when models are based exclusively on ber of models for slowly rotating asteroids, until recently thermal dense lightcurves, the uncertainty of the spin axis position (pole) inertia values from detailed TPM have been determined for only was of the order of a few degrees (see Table1), and the shape two such asteroids (227 Elvira and 956 Elisa, Delbo’ & Tanga models had a smooth appearance. Formally large uncertainty in 2009; Lim et al. 2011; see Fig. 5 in Harris & Drube 2016; based some values of pole longitude (λp) is due to high inclination of on compilation by Delbo’ et al. 2015, reproduced here in Fig. 12) the pole (βp), translating actually to a relatively small distance The above-mentioned trend was found on the values derived on the celestial sphere. from estimated beaming parameters using a simplified approach. In thermophysical modelling, the diameter (D) and the ther- Recent results from TPM by Hanuš et al.(2018a) showed a rather mal inertia (Γ) are fitted and different surface roughness are tried large diversity of thermal inertia values for slow rotators within by varying the opening angle of hemispherical craters cover- the size range of 10–100 km. Here we further investigate this ing 0.6 of the area of the facets (following Lagerros 1996), issue by studying more slow rotators in both visible light and covering rms values between 0.2 and ∼1. Heat diffusion is 1D thermal infrared radiation. and we use the Lagerros approximation (Lagerros 1996, 1998; Our photometric campaign is ongoing, and new observing Müller & Lagerros 1998; Müller 2002). The TPM implemen- stations have joined. Our data sources now also include observa- tation is the one used in Alí-Lagoa et al.(2014), based on that tions from the Kepler Space Telescope in its extended mission, of Delbo’ & Harris(2002), however the colour correction is K2 (Howell et al. 2014). Kepler could not track moving targets treated differently: instead of colour correcting each facet’s flux during a campaign, so solar system objects could be observed based on its effective temperature, we now use the H and G either by using previously allocated, curved, or boomerang- values tabulated in the JPL (Jet Propulsion Laboratory) Hori- shaped masks in accordance with the expected positions of the zons database to compute an effective temperature based on the target minor bodies (see e.g. Pál et al. 2015) or larger continu- heliocentric distance at which each observation was taken (fol- ous pixel masks (so-called supermasks) where such objects also lowing the approach in Usui et al. 2011). Another simplifying appeared for several days, but not on purpose (see e.g. Molnár assumption is that spectral emissivity is constant and equal to et al. 2018). Data processing has been safely established for 0.9 (see e.g. Delbo’ et al. 2015), which does not seem unreason- these observations, regardless of whether these were targeted or able given the small spectral contrast of 1–3% found by Licandro not. This processing is based on the registration of the images et al.(2012) in the 10- µm features of 50–60 km Themis family (to correct for spacecraft jitter) followed by a differential-image members (the 20-µ emission plateau is expected to be flatter). photometry by involving oblong-shaped apertures (see Pál et al. Once we obtain D, we compute the visible geometric albedo 2015; Szabó et al. 2017 and Molnár et al. 2018 for further details). using the H-G12 values from Oszkiewicz et al.(2011). The Bond The model of (100) Hekate presented here has been partially albedo that would be derived from these values is sometimes dif- based on Kepler data. The asteroid (100) Hekate was observed ferent from the Bond albedo value used as input for the TPM with a dedicated set of masks for 5.5 and 4.5 days continuously (Ai). This value was obtained by first averaging the tabulated in Campaigns 16 and 18, respectively. Table A.1 gives details of diameter from IRAS, AKARI, and WISE (Tedesco et al. 2005; each observing run obtained within the present work. Scarce lit- Usui et al. 2011; Alí-Lagoa et al. 2018; Mainzer et al. 2016) and erature data had to be complemented with from ∼100 up to ∼500 occultations (compiled by Dunham et al. 2016). Then, we used h of new observations for each target before unique models were this size to compute the Bond albedo from all available H-G, feasible. H-G12, and/or H-G1-G2 values from the Minor Planet Center

A139, page 3 of 40 A&A 625, A139 (2019)

Table 1. Spin parameters of asteroid models obtained in this work, with their uncertainty values.

Sidereal Pole 1 Pole 2 rmsd Observing span Napp Nlc

period (h) λp βp λp βp (mag) (yr) (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◦

Notes. 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.

(Oszkiewicz et al. 2011 or Vereš et al. 2015), and took again the like Fig. B.9 show the observation-to-model ratios versus wave- average value. This approach is somewhat arbitrary but the TPM length, helicentric distance, rotational angle, and phase angle. results are not very sensitive to the value of the Bond albedo. Table1 presents spin parameters obtained within this work Justification for this, further details, and discussion, including (sidereal period, spin axis position, and used lightcurve data). information on how we estimate our error bars, are given in Results from thermophysical modelling with our spin and shape AppendixB. models applied are summarised in Table2, presenting the diam- eter, albedo, and thermal inertia for the best-fitting model solu- 4. Results tion. The values obtained here for diameter are “scaling values” of the given spin and shape solutions listed in Table 1 that would In this section for each target separately we refer to previous otherwise be scale-free. For comparison we also refer there to works containing lightcurves, briefly describe our data, the char- the diameters obtained previously from the AKARI (Usui et al. acter of brightness variations, and its implications for the spin 2011), IRAS (Tedesco et al. 2005), and WISE (Mainzer et al. and shape. Later, the thermal data availability and range are 2011; Masiero et al. 2011) surveys. We also added their tax- described, followed by the results of applying these models in onomic class following Bus & Binzel(2002a,b) and Tholen TPM analysis. In plots like Fig.1, the fit of model infrared (1989), so that the albedos could be verified for agreement with flux variations to thermal lightcurves is presented, and plots taxonomy.

A139, page 4 of 40 A. 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).

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

Notes. 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.

10.5 AM 2 model evolution caused by phase angle effect, which due to various Abs.calib. error bar 10 levels of shadowing highlight various shape features. Since we applied no photometric phase correction, their signatures are 9.5 visible on the overlapping fragments that cover the same rota- 9 tional phase, but are spaced by more than a month in time like

8.5 for example in Fig. A.18. Apparitions with only one lightcurve

Flux (Jy) fragment or those with data covering only a small fraction of 8 full rotation, are not presented here, because creating compos- 7.5 ite lightcurves would be either impossible or would not present

7 much information on the period or the lightcurve appearance. However, such data have been used in the modelling, helping 6.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 to constrain spin and shape. The full list of observing runs is Rotational phase summarised in Table A.1.

Fig. 1. (100) Hekate W4 data and model thermal lightcurves for shape 4.1. (100) Hekate model 2. Data error bars are 1σ. Table B.1 summarises the TPM analysis. We compiled archival lightcurves from four apparitions, pub- lished by Tedesco(1979), Gil-Hutton(1990), Hainaut-Rouelle et al.(1995), and Galad et al.(2009), and complemented them For practical reasons, the projections of the shape models with data from the SuperWASP survey (Wide Angle Search for and the fit to all the visible lightcurves are not presented here, but Planets, Grice et al. 2017), and our own data from three more will be available from the DAMIT (Database of Astroid Models apparitions (see Figs. A.1–A.4). SuperWASP cameras are known from Inversion Techniques)2 created by Durechˇ et al.(2010), for to suffer from the detector having a temperature dependence, so models viewing and download, to be used in other applications. the absolute lightcurves can be systematically shifted in magni- In AppendixA (from Figs. A.1 to A.53), we present all the pre- tude depending on the temperature of the detector. However, as viously unpublished photometric data in the form of composite we treat them as relative lightcurves this should not be an issue. lightcurves, as they testify the good quality of our spin and shape In the last apparition, Hekate was observed for our project by solutions: photometric accuracy here is mostly at the level of a the Kepler Space Telescope, resulting in two continuous, four- few millimagnitudes, which is one to two orders of magnitude day-long lightcurves of great quality. The K2 data have been better than in sparse data standardly used for asteroid modelling. reduced with the fitsh package, using the same methods that Also, with such slow rotation, separate lightcurves would not were already applied to targeted observations of Trojan asteroids show the whole character of the brightness variability, but only a and chance observations of main-belt asteroids (MBAs) in the small fraction of the full lightcurve, which is visible only in the mission (Pál 2012; Szabó et al. 2016, 2017; Molnár et al. 2018). folded plots. Moreover, such plots enable us to see lightcurve Overall, Hekate displayed interesting, asymmetric lightcurves of amplitudes ranging from 0.11 to 0.23 magnitude and a long, 2 http://astro.troja.mff.cuni.cz/projects/asteroids3D 27.07 h period. A139, page 5 of 40 A&A 625, A139 (2019)

2.2 The spin parameters of our model are presented in Table1. AM 1 model To construct it, initial scanning of parameter space needed an Abs.calib. error bar increased number of trial poles and iteration steps. Overall both 2 spin solutions fit the visual lightcurves at a very good level of

0.012 mag. 1.8

TPM analysis Flux (Jy) 1.6 The two mirror pole solutions of Hekate are named AM 1 and AM 2. We used 52 infrared observations, 32 from IRAS 1.4 (8 × 12 µm, 8 × 25 µm, 8 × 60 µm, 8 × 100 µm), five from 1.2 AKARI (2 × S9W, 3 × L18W), and 15 from WISE (W4). We 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 assumed A = 0.090 for the Bond albedo. Rotational phase Both models provide formally acceptable fits, as do the cor- 5 AM 1 model 2 Abs.calib. error bar responding spheres (see Table B.1 and the χ plots in Fig. B.9). 4.8 The fit to the WISE “lightcurve” seems reasonable for both mod- els, although not around the zero rotational phases shown in 4.6 Figs.1, B.1, and B.9 (third panel from the top). The IRAS data 4.4 present a slope in the observation-to-model ratios (OMR) ver- 4.2

sus wavelength plot, which is shown in the top panel of Fig. B.9. Flux (Jy) 4 Model 2 provides a better fit than 1 and the spheres so we select it as our provisionally favoured solution. Nonetheless, these results 3.8 could be biased by our neglecting the dependence of thermal 3.6 conductivity – and therefore thermal inertia – with temperature 3.4

(see Rozitis et al. 2018, and the review in Delbo’ et al. 2015, for 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 instance), over the wide range of heliocentric distances sampled Rotational phase (2.6–3.6 au). However, there is not enough data taken at long heliocentric distances to make a conclusive statement. Fig. 2. Asteroid (109) Felicitas W3 (top) and W4 (bottom) data and To summarise the TPM analysis: the surface roughness is model thermal light curves for shape model AM 1 (see Table B.2). +5 not constrained, the diameter is 87−4 km, and geometric albedo, p = +0.03 V 0.22−0.03. We would benefit from additional thermal light Even though the model thermal lightcurves miss some W3 and curves at negative phase angles (i.e. after opposition) and W4 fluxes (Fig.2), we consider it as a preliminary approx- higher heliocentric distances (to study the conductivity depen- imated solution given that it fits significantly better than the dence with temperatures). Low thermal inertias of approximately 3 sphere. Additional thermal curves could help confirm or reject 5 SI units and low to medium roughness fit the data better. this model and data at negative phase angles could improve the constraints for the thermal inertia. 4.2. (109) Felicitas As usual, the surface roughness is not constrained. From the diameter (85+7 km), we obtain the albedo p = 0.065+0.008. Available photometric data for Felicitas came from the works −5 V −0.01 of Zappala et al.(1983), Harris & Young(1989), and from the SuperWASP archive (Grice et al. 2017). We added to these two 4.3. (195) Eurykleia apparitions data from five more obtained within our campaign, There was no prior publication presenting lightcurves of Euryk- and also data from the WISE satellite obtained in W1 band, leia, but plots from three apparitions are available on the which is dominated by reflected light (thermal contribution in observers’ web pages4, indicating a long, 16.52 h period. Here we W1 for this target was estimated at 8–30%). The latter dataset publish all of these data, adding more recent observations from greatly helped to constrain the model, in an approach first pro- four more apparitions. During all the seven apparitions, Euryk- posed by Durechˇ et al.(2018). Felicitas lightcurves have been leia displayed similarly shaped lightcurves with one minimum changing substantially between the apparitions, reaching peak- sharper than the other, which was accompanied by an additional to-peak amplitudes from 0.06 to 0.22 mag, with the synodic bump or shelf (see Figs. A.9–A.14). The amplitude was stable, period around 13.194 h (Figs. A.6–A.8). being always around 0.24 mag, which indicates a stable aspect angle due to the high inclination of the spin axis. As expected, TPM analysis the resulting model has a high value of |β|, and a formally large range of possible values of λ (see Table1). The two mirror solutions for Felicitas from Table1 are denoted AM 1 and AM 2. In the thermal approach we used 38 obser- vations, 15 from IRAS (5 × 12 µm, 7 × 25 µm, 3 × 60 µm), five TPM analysis from AKARI (3 × S9W, 2 × L18W), and 18 from WISE (9 × W3, The two mirror solutions from lightcurve inversion are labelled 9 × W4). We assumed A = 0.025 for the Bond albedo. AM 1 and AM 2. We used a total of 57 infrared observations Unlike the AM 2 model, AM 1 provides a formally accept- from IRAS (nine epochs × four filters), AKARI (4 S9W + 5 able fit (Table B.2, and the χ2 plot in Fig. B.10), although the L18W) and WISE W4 (12). The WISE W3 data were satu- WISE bands are not fitted equally well: the model overestimates rated for this object and the W4 data, with reported magnitudes the W4 data and underestimates the W3 (top panel in Fig B.10). 4 http://obswww.unige.ch/~behrend/page_cou.html, 3 (Jm−2s−0.5K−1). http://eoni.com/∼garlitzj/Period.htm

A139, page 6 of 40 A. Marciniak et al.: Thermal properties of slowly rotating asteroids

9.5 4.2 AM 1 model W4 data Abs.calib. error bar 9 Best fit AM 1 (all data used) Best fit AM 1 (W4 not used) Best fit AM 1 (only W4 used) 4 8.5

8 3.8

7.5

3.6 7 Flux (Jy) Flux (Jy)

6.5 3.4 6

5.5 3.2

5 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. 3. W4 data and model thermal lightcurves for (195) Eurykleia’s Fig. 4. Asteroid (301) Bavaria AM 1 model thermal lightcurves and W4 shape model AM 1. The different models resulted from fitting different data (see Table B.4). subsets of data. Table B.3 contains the corresponding thermo-physical parameters. roughness models fit the data better, as is usually the case for main-belt asteroids. between 0.00 and 0.35, are near the −0.6 mag identified as the Finally, the geometric albedos derived from our radiometric onset of saturation for individual W4 images (Cutri et al. 2012). diameters vary slightly depending on the photometric phase cor- We assumed A = 0.020 for the Bond albedo. rection applied and the source of the absolute magnitudes and Both shape models provide statistically similarly good fits to slope parameters. To compute our final value, we use the H-G12 all the data, so the TPM cannot reject any of the mirror solu- values from the Oszkiewicz et al.(2011) table and calculate the tions in this case. Using all the data to optimise the χ2 (i.e. maximum and minimum possible pV s using all the three sources considering ν = 57 − 2 = 55 degrees of freedom), the minimum p = . ± . 2 listed above. We find V 0 06 0 02. reduced chi-squared (χ¯ m) were 0.51 and 0.60 for the AM 1 and AM 2 models, respectively. These are significantly better than the values obtained for spheres with the same respective spin 4.4. (301) Bavaria pole orientation (see Table B.3). The lightcurve from only one previous apparition of asteroid Although the thermal data seems to be fitted very well based Bavaria has been reported in the literature (Warner 2004). Our 2 on our small χ¯ m, the OMR of many IRAS data are slightly but campaign built an extensive lightcurve dataset for this target, systematically above 1, the WISE OMRs slightly below 1, and covering four viewing aspects and a wide range of phase angles. the AKARI ones are not fully horizontally aligned (Fig. B.11, Our data covered sometimes four to five months within one upper panels). Figures3 and B.2 show that both shape models fit apparition, and less than a year passed between consecutive the variation of the W4 thermal lightcurve reasonably well. How- observed apparitions. Such an observing strategy was defined ever, the fluxes predicted by the TPM solution that best fits all as the most optimal for spin and shape reconstructions by the data (yellow open circles) systematically overestimate most Slivan(2012), and is confirmed by our experience. Bavaria of the W4 observations. This is worse for the solution that best during its 12.24-h-long rotation displayed wide minima and fits the AKARI and the IRAS data (open triangles), because the amplitudes from 0.25 to 0.36 magnitudes (Figs. A.15–A.18), fitted diameter is larger in this case (third line in Table B.3, exceeding our initial target selection criteria in the course of the first and third plot from the top in Fig. B.12). However, the campaign; nonetheless it was retained in the target list. Although OMR values for the IRAS and AKARI data do align horizon- the solution for the sidereal period and two possible spin axis tally (Fig. B.11, lower panels). Finally, we can fit the lightcurve positions are very well defined, due to the high value of pole much better if we optimise the W4 data alone (filled squares), latitude (see Table1) the shape model vertical extent is poorly but the thermal inertia is lower by a factor of three (fourth entry constrained. in Table B.3, second and fourth plot in Fig. B.12). Conclusions from TPM analysis are the following: both TPM analysis shapes seem to fit the data well, including the shape of the W4 thermal lightcurve (Fig.3), although the flux level of the W4 The two mirror solutions from Table1 are AM 1 and AM 2. data is systematically below our fitted model’s fluxes. We do not We used 36 thermal observations, 18 from IRAS (9 × 25 µm, have an explanation for such a systematic mismatch, but perhaps 9 × 60 µm), six from AKARI (3 × S9W, 3 × L18W), and 12 from additional thermal IR data could correct the TPM diameter of WISE (W4). We rejected IRAS 12- and 100-micron data because (195) Eurykleia to a higher value closer to 90 km, or a lower they contain clear outliers (by a factor of several). We assumed value closer to 80 km. Thus, with the current dataset, the possible A = 0.020 for the Bond albedo. diameters range from 78 to 98 km depending on which subsets Both models and the corresponding spheres provide a good of data we fit (Table B.3). As a compromise, we take the mid- fit (see Table B.4, and the χ2 plots in Fig. B.13), although with +11 ∼ value in this range, 87−9 km (3σ level error bars, 13% relative slightly offset diameters and different roughness (the spheres error). require low roughness rather than medium). The AM 1 OMR On the other hand, the large 3σ range of possible thermal plots are shown in Fig. B.13, and the W4 model thermal inertias does not change when we fit different subsets of data, lightcurves with the data in Figs.4 and B.3. but the error bars are widely asymmetric. We can only provide As usual, the roughness is not constrained at the 3σ level. +2 +0.004 an upper limit of 70 SIu, with a best-fitting value of Γ = 15 SIu. With a diameter 55−2 km, we get pV = 0.047−0.003 and a thermal Surface roughness is not constrained at the 3σ level, but high inertia between 10 and 100 SI units.

A139, page 7 of 40 A&A 625, A139 (2019)

10 9 W4 data W4 data Best fit AM 2 Best fit AM 2 8 9 7

8 6 Flux (Jy) Flux (Jy) 5 7

4 6 3

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. 5. Asteroid (335) Roberta’s WISE data and model thermal Fig. 6. Asteroid (380) Fiducia’s WISE data and model thermal lightcurves for shape model 2 and the corresponding best-fitting solu- lightcurves for shape model 2 and the corresponding best-fitting solu- tions (very low thermal inertia of 15 SIu). tions (very low thermal inertia of 10 SIu; see Table B.6).

4.5. (335) Roberta exclusively on the 0.8m TJO (Telescopi Joan Oró) telescope in the Montsec Observatory in a mode similar to “dense-sparse We compiled a large dataset covering as many as nine appari- cadence” described by Warner & Harris(2011). In spite of the tions of asteroid Roberta, including data from Binzel(1987), relative sparseness of datapoints, the lightcurve’s general char- Harris et al.(1992), Lagerkvist et al.(1995), Warner et al.(2007), acter is clearly outlined, and the synodic period from other Pilcher & Martinez(2015), SuperWASP archive (Grice et al. apparitions is confirmed (Fig. A.24). 2017), and our own data (presented in Figs. A.19–A.21). This In the apparition on the verge of the years 2015 and 2016, target showed rather rare, monomodal lightcurves of amplitudes Fiducia was observed under an exceptionally large range of phase from 0.13 to 0.19 mag, depending on the aspect. Its rotation angles (Table A.1), which resulted in distinctive differences in period, 12.027 h, commensurate with an Earth day, required the lightcurve character. Thus we present data from that appari- observations from sites well spaced in longitude for full cover- tion on two separate plots, one for small, and the other for large age. Thanks to such an extensive dataset both spin and shape phase angles (Figs. A.26 and A.25, respectively). solutions are well constrained (Table1), however different from The unique solution for the sidereal period could only be those found by Blanco et al.(2000), who reported four pole solu- found in this case during denser scanning of parameter space. tions, all much lower in |β| than ours, and a sidereal period longer In spite of clear signatures of low inclination of the spin axis, the by 0.027 h, which is a substantial difference. resulting pole solution is located at rather moderate latitudes (see Table1). This might be a symptom of the lightcurve inversion TPM analysis method bias against low poles found by Cibulková et al.(2016), as small shape modifications can compensate for the shifted spin The two mirror solutions are AM 1 and AM 2. Thermal axis position of the model. data consisted of 44 observations, 22 from IRAS (6 × 12 µm, × µ × µ × µ × 6 25 m, 6 60 m, 4 100 m), nine from AKARI (4 S9W TPM analysis and 5 × L18W), and 13 from WISE (W4). We assumed A = 0.020 for the Bond albedo. The two obtained mirror solutions are AM 1 and AM 2. We The best solutions for both models are comparably good, and had 54 thermal observations at our disposal, 38 from IRAS even the spheres provide a formally good fit with compatible (10 × 12 µm, 10 × 25 µm, 10 × 60 µm, 8 × 100 µm), seven from results (see Table B.5, and the χ2, and OMR plots in Fig. B.14). AKARI (5 × S9W and 2 × L18W), and nine from WISE (W4). The diameter is constrained at the 3σ-level but with a relatively We assumed A = 0.020 for the Bond albedo. ∼ 2 large relative error of 10%. The fit to the WISE lightcurve The best solutions for both models have comparable χ¯ min and (Figs.5 and B.4) shows that the fit at phases between 0.0 and 0.10 2 very shallow minima in the χ¯ min versus Γ plots. Thermal iner- might be improved with additional visible data for the inversion tias between 0 and 150 SIu fit the data (3σ limits), but the best (phase 0 corresponds to JD 2442489.835789). fit is Γ = 10 SI units (see Table B.6, the χ2 plot, and OMR plot 2 The problem with this model is that the χ¯ min versus Γ curve in Fig. B.15). The corresponding spheres give similar diameters is very shallow at high Γs, so the thermal inertia is basically but fit the data better with higher thermal inertia (Γ ≈200 SIu). unconstrained. The aspect angles sampled by the thermal data Only the most extreme roughness values produce fits outside are concentrated around equatorial values, so additional data at the 3σ range, and low roughness solutions (rms < 0.30) fit the other sub-observer latitudes and a dense thermal lightcurve at data better. The shapes reasonably reproduce the W4 lightcurve pre-opposition (positive phase angle) would help. The diameter (Figs.6 and B.5), but the fits could probably be improved with +10 +0.014 constraint of 98−11 km leads to an albedo of pV = 0.046−0.008. “less boxy” shape models. From TPM analysis one can conclude that the roughness is 4.6. (380) Fiducia unconstrained, but thermal inertias are lower than 150 at the 3σ level (the best fit is for Γ = 10 SIu). The diameter range of A previous lightcurve of Fiducia was published by Warner +9 +0.009 72 km leads to an albedo pV = 0.057 . (2004). With this work we add six apparitions, confirming the −5 −0.012 synodic period around 13.72 h. Fiducia’s lightcurve shape varied 4.7. (468) Lina substantially from one apparition to another, reaching 0.32 mag amplitude in one, and then decreasing down to 0.04 in the other Lightcurves from two previous apparitions of Lina have been (see Figs. A.22–A.28). In the year 2014 Fiducia was observed published by Tedesco(1979) and Buchheim(2007). Our visual A139, page 8 of 40 A. Marciniak et al.: Thermal properties of slowly rotating asteroids

8 W4 data TPM analysis Best fit AM 1 ◦ ◦ 7 The two mirror spin-shape solutions are AM 1 (168 ,−58 , 46.73928 h) and AM 2 (328◦,−59◦, 46.73985 h). Friederike is

6 well observed at thermal wavelengths: two four-band IRAS mea- surements, ten measurements by AKARI (4 × S9W, 6 × L18W),

Flux (Jy) 5 taken at different epochs before and after opposition, and 23 WISE data points (at W3 and W4), also at two epochs before

4 and after opposition. Solution AM 1 provides a borderline acceptable fit to all ther-

3 mal data simultaneously, as do the corresponding spheres (see 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Table B.8). However, the AM 2 solution seems to match the Rotational phase WISE W3 and W4 lightcurves a bit better. The residuals in the Fig. 7. Asteroid (468) Lina’s WISE data and model thermal lightcurves observation-divided-by-model plots (see Fig. B.17) indicate that for shape model 1’s corresponding best-fitting solutions (see Table B.7). the spin-shape solutions are not perfect or – alternatively – that there are surface variegations that influence the thermal fluxes. The WISE W3 and W4 lightcurves favour AM 2, but do not help observations of this target spanned six apparitions, and we found to settle the spin ambiguity completely. Overall, in this case high it displaying complex lightcurves, with wide, wavy maxima and extremely high surface roughness worked very well and all and narrow minima (Figs. A.29–A.33). Peak-to-peak amplitudes the radiometric solutions point to low values for the thermal iner- ranged between 0.13 and 0.18 mag within its 16.48 h period. tia well below Γ = 20 SIunits (see χ2 plots in Fig. B.17), with a Model spin parameters are given in Table1. Shape model stretch trend to higher inertias at shorter heliocentric distance (2.6 au) along the spin axis is somewhat uncertain. and lower values at the largest heliocentric distance (>3.5 au). The heliocentric influence on thermal inertia is also visible in the TPM analysis radiometric solutions for the individual datasets (see Table B.8): all WISE data are taken at rhelio > 3.4 au, while the AKARI and The two mirror solutions are named AM 1 and AM 2. We IRAS data are taken well below 3.0 au. The overall best radio- × × used 33 observations, seven from IRAS (2 12 µm, 2 25 µm, metric solution for AM 2 and extremely high surface roughness × × × 2 60 µm, 1 100 µm), eight from AKARI (2 S9W and produces Γ = 20 SIunits, a diameter of 76+4 , and a geometric × −2 6 L18W), and 20 from WISE (W4), and assumed A = 0.020 albedo of 0.06 ± 0.01. for the Bond albedo. The comparatively fewer data points are available for this 4.9. (653) Berenike object. The best-fitting roughness and thermal inertias are low (rms ∼ 0.3 and Γ = 20 SIu; see Table B.7, and the χ2 plots in Berenike displayed small amplitudes in the range of 0.05–0.16 Fig. B.16), even for the spherical models. The fit to the WISE (Figs. A.38–A.41), also in the archival observations by Binzel 2 lightcurve is reasonably good (Figs.7 and B.6), but the χ¯ min (1987) and Galad & Kornos(2008). However, in the last appari- is 1.20, which is not optimal. The data sample widely differ- tion observed in this work the amplitude unexpectedly rose to ent heliocentric distances, from 2.5 to 3.5 au (Fig. B.16), but the 0.38 mag, being over two times larger than ever, which is con- residuals do not present a strong trend. firmed by two independent observing runs (Fig. A.42). That The analysis points to thermal inertias lower than 300 SIu apparition must have provided the only viewing geometry when and low roughness, but this object requires more thermal data signatures of full elongation of this target could be visible. This to constrain these properties better. Even the diameter has a case shows the importance of probing a wide range of geome- +11 relatively large error bar by TPM standards: 69−4 km. The tries for correct reproduction of the spin axis position and shape +0.006 corresponding albedo is 0.0520.014 . elongation. The synodic period was around 12.485 h, and the full dataset spanned seven apparitions. 4.8. (538) Friederike The shape model is indeed elongated in the equatorial dimen- sions, and this time the pole position is low, as expected (see Data previously published came from one apparition and clearly Table1). Still, the shape model extent along the spin axis displayed a very long, 46.728 h period for this target (Pilcher is poorly constrained, probably due to the lack of data from 2013). In addition to it we used previously unpublished data geometries where intermediate amplitudes could be observed. from the years 2003 and 2006, with only partial coverage though (Table A.1), and own data from five more apparitions. TPM analysis Those with good phase coverage are shown in Figs. A.34–A.37. Lightcurves of Friederike had minima of different widths and We denote the two mirror solutions as AM 1 and AM 2. We used amplitudes from 0.20 to 0.25 mag. 49 thermal observations, 24 from IRAS (8 × 12 µm, 8 × 25 µm, In spite of a large and varied dataset, only with the addition 8 × 60 µm), eight from AKARI (4 × S9W, 4 × L18W), and 17 of WISE W1 data could the unique solution for period, spin axis, from WISE (8 × W3, 9 × W4). The Bond albedo was assumed and shape be found. These data proved to be a necessary, contin- to be A = 0.070. uous basis lasting a few tens of hours, while other lightcurves Model AM 1 is significantly better than AM 2 and the covered only at most 20% of the full rotation. Such extremely spheres in this case, although the reduced χ2 is 1.1, slightly over long period targets are especially challenging for ground-based unity (Table B.9, and the χ2 plot in Fig. B.18). The IRAS data studies, being better targets for wide-field, space-borne obser- residuals show some scatter and the WISE data are reasonably vatories. Due to this long period and relatively short time span well fitted but still present some waviness in the OMR versus of the observational dataset, the precision of the sidereal period rotational phase plot (Figs.9 and B.18). The latter could indicate determination is lower than in the case of other targets (Table1). that there is room for improvement of the shape model. Very high A139, page 9 of 40 A&A 625, A139 (2019)

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

0.72 0.65 0.7 Flux (Jy) 0.68

Flux (Jy) 0.60

0.66 0.55 0.64

0.50 0.62

0.6 0.45 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 Rotational phase 2.05 AM 1 model Rotational phase Abs.calib. error bar 2 1.15 AM 2 model 1.95 1.10 Abs.calib. error bar 1.9 1.05 1.85

1.00 1.8

0.95 Flux (Jy) 1.75

0.90 1.7

Flux (Jy) 1.65 0.85 1.6 0.80 1.55 0.75 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.70 Rotational phase 0.65 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 9. Asteroid (653) Berenike WISE data (W3 top, W4 bottom) and Rotational phase the AM 1 model’s thermal lightcurves (see Table B.9). 2.40 AM 2 model Abs.calib. error bar 2.20 4.10. (673) Edda

2.00 The first published lightcurve of Edda was obtained within our project (Marciniak et al. 2016), and was highly asymmetric, with

Flux (Jy) maxima unequally spaced in time. Here we present data from 1.80 five more apparitions, confirming the period of 22.34 h, and non- typical, asymmetric lightcurve behaviour (see Figs. A.43–A.47). 1.60 This dataset was complemented by one more apparition with par- tial coverage by data from the SuperWASP archive (Grice et al. 1.40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2017). lightcurve amplitudes ranged from 0.13 to 0.23 mag. The Rotational phase shape model from lightcurve inversion is somewhat angular, and the fit to some of the lightcurves of smallest amplitude is imper- 3.00 AM 2 model Abs.calib. error bar fect. Both spin solutions are well constrained and are given in Table1. 2.80

2.60 TPM analysis

Flux (Jy) 2.40 The two mirror solutions are named AM 1 and AM 2. At our disposal there were 54 infrared observations, 20 from IRAS 2.20 (7 × 12 µm, 7 × 25 µm, 6 × 60 µm), six from AKARI (3 × S9W and 3 × L18W), and 28 WISE (14 × W3 and 14 × W4). We 2.00 assumed A = 0.047 for the Bond albedo. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 The best solutions for both models have comparable and very Rotational phase 2 low χ¯ m with very low thermal inertia of about 3 SI units (see 2 2 first two plots from the Table B.10 and the χ plots in Fig. B.19). The low χ¯ ms suggest the Fig. 8. Asteroid (538) Friederike’s WISE W3 ( 2 top) and W4 data (last two plots) taken at two epochs, and thermal error bars might be overestimated, so we normalise the χ¯ curves lightcurves for shape model AM 2 and the corresponding best-fitting to have the minimum at 1 in order to compute the uncertainties of solutions obtained based on all thermal data. the parameters (see discussion in Hanuš et al. 2015). The shapes fit the WISE data well (Figs. 10 and B.7), and unlike (195) Euryk- roughness solutions fit the data better, and an rms lower than 0.3 leia’s case, the best-fitting solution also fits the AKARI data with can be rejected at the 3σ level. a reasonably flat OMR versus wavelength plot (Fig. B.19). We conclude that additional densely sampled thermal To conclude, although high thermal inertias of about 70 lightcurves and an improved shape model could improve the SI units are still allowed, the best solutions seem to point to constraints on the thermal inertia and surface roughness. With very low thermal inertias of around 3 SI units. The diame- +4 +0.02 ters are constrained to be 38+6 km (3σ error bars), which lead a diameter 46−2 km, we get pV = 0.18−0.03, and the best fit is −2 +0.03 obtained for medium values of thermal inertia. to a visible geometric albedo of pV = 0.13−0.05. This value is A139, page 10 of 40 A. Marciniak et al.: Thermal properties of slowly rotating asteroids

0.75 W3 data W3 data Best fit AM 1 1.4 Best fit AM 1 0.7

1.2 0.65

1 0.6 Flux (Jy) Flux (Jy)

0.55 0.8

0.5 0.6

0.45 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 Rotational phase Rotational phase W4 data 1.7 3.4 Best fit AM 1 W4 data Best fit AM 1 1.6 3.2

3 1.5 2.8 1.4

Flux (Jy) 2.6

Flux (Jy) 1.3 2.4 1.2 2.2

1.1 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 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. 11. Asteroid (834) Burnhamia’s WISE data and model thermal Fig. 10. Asteroid (673) Edda’s WISE data and model thermal lightcurves for shape model 1’s best-fitting solution (see Table B.11). lightcurves for shape model 1’s best-fitting solution (very low thermal inertia of 3 SI units). 5. Discussion and conclusions We present here 11 new spin and shape models of slowly rotating somewhat low for a typical S-type (classified based on a visi- asteroids from lightcurve inversion on dense lightcurves, with ble SMASS spectrum, Small Main-Belt Asteroid Spectroscopic determined sizes, albedos, and thermal parameters. Together Survey Bus & Binzel 2002a,b). with five models from our previous study (Marciniak et al. 2018), we have 16 shape models within our sample of slow-rotators 4.11. (834) Burnhamia with applied TPM. This number substantially enlarges the avail- able pool of well-studied slow rotators. Models obtained in this Previously observed by Buchheim(2007) and in our cam- work provide good resolution, “smooth” shape representations paign, Burnhamia displayed lightcurves of extrema at unequal for a multitude of future applications, including calibration in levels, 13.87 h period, and amplitudes from 0.15 to 0.25 the infrared. mag. We accumulated data from six apparitions well spread One of our aims was to verify whether the thermal iner- in longitude, and present them in Figs. A.48–A.53. In the tia indeed increases with the rotation period (Harris & Drube lightcurve inversion we obtained two well-defined pole solutions 2016). Our initial sample (Marciniak et al. 2018) seemed to be (Table1) with somewhat unconstrained shape models both in in line with this finding, while our current sample is dominated the vertical dimension and in the level of smoothness of the by targets with lower thermal inertias. To ensure this different shape. result is not related to the fact that we used a different code for this work5, we modelled one of the targets from Marciniak TPM analysis et al.(2018) and reproduced the results. This is expected given The two mirror solutions are denoted AM 1 and AM 2. We used that we use the same model approach and approximations. We 70 thermal observations, 42 from IRAS (11 × 12 µm, 11 × 25 µm, also cross-checked our results with those estimated by Harris & 11 × 60 µm, 9 × 100 µm), one from AKARI (L18W), and 26 from Drube(2016) and found one overlapping target, 487 Venetia, WISE (13 × W3, 13 × W4). We assumed A = 0.035 for the Bond with consistent thermal inertia values: 96 SIu in Harris & Drube albedo. (2016), and 100 ± 75 SIu in Marciniak et al.(2018). In Fig. 12 It turned out to be yet another slow rotator with seemingly we superimposed our results on the plot from Harris & Drube low thermal inertia. The three-sigma upper limit on 50 SI is (2016). quite clear. The χ¯ 2 is ∼0.8 for both models (Table B.11, and Taking both of our samples together (from this work and min from Marciniak et al. 2018), we find diverse values of thermal the χ2 plots in Fig. B.20), so the fit is good but there is room for inertia, ranging from 3 up to 125 SI units. To make a quantita- improvement. The WISE data are reasonably well fitted (Fig.11, tive comparison, we collected the thermal inertias of all targets and B.8), but the 100 µm IRAS data are not (these data carry less with sizes 30 < D < 200 km from Hanuš et al.(2018a), Delbo’ weigh for the fit, however). et al.(2015), and our values and split them into two groups: slow As usual, the roughness is not constrained, the diameter is +8 +0.014 (P > 12 h) and fast (P < 12 h) rotators. Neither the two-sample 67−6 km and pV = 0.074−0.016. We could benefit from additional thermal light curves at positive phase angles (pre-opposition) 5 We used the TPM code of Müller(2002) in Marciniak et al.(2018), and lower heliocentric distance although there is no apparent whereas here we used the TPM code of Delbo’ & Harris(2002) trend in the lowest panel of Fig. B.20. modified in Alí-Lagoa et al.(2014) mentioned in Sect.3. A139, page 11 of 40 A&A 625, A139 (2019)

frames are going to be downlinked, so TESS data are going 600 to be perfect for extensive studies of slow rotators (Pál et al. 2018). It is also possible that the shape models that best fit visual 500 lightcurves were not the best possible ones from the thermal point of view, so the varied-shape TPM method (Hanuš et al. 400 2015), or simultaneous fitting of visual and thermal data using a convex inversion TPM (Durechˇ et al. 2017) could help to resolve 300 issues with unconstrained solutions. To overcome problems with limited thermal data we are planning proposals to VLT/VISIR

Thermal inertia [SI units] 200 (Very Large Telescope with the VLT spectrometer and imager for the mid-infrared) and SOFIA (Stratospheric Observatory For 100 Infrared Astronomy) infrared facilities, carefully choosing tar- gets and observing geometries to best complement the existing 0 datasets in terms of aspect, pre- and post-opposition geometry, 10 100 Rotation period [h] and heliocentric distance.

Fig. 12. Updated plot from Harris & Drube(2016) of thermal inertia Acknowledgements. This work was supported by the National Science Centre, dependence on rotation period. Black dots are thermal inertias estimated Poland, through grant no. 2014/13/D/ST9/01818. The research leading to these from η values by Harris & Drube(2016), red dots from detailed TPM results has received funding from the European Union’s Horizon 2020 Research compiled by Delbo’ et al.(2015), orange are slow rotators from “Varied- and Innovation Programme, under Grant Agreement no 687378 (SBNAF). The research of V.K. was supported by a grant from the Slovak Research and Shape TPM” by Hanuš et al.(2018a), blue are from TPM results of Development Agency, number APVV-15-0458. R.D. acknowledges financial Marciniak et al.(2018), and green dots from this work. Our new results support from the State Agency for Research of the Spanish MCIU through the do not confirm the growing trend. “Center of Excellence Severo Ochoa” award for the Instituto de Astrofísica de Andalucía(SEV-2017-0709). The Joan Oró Telescope (TJO) of the Montsec Kolmogorov-Smirnov test nor the k-sample Anderson-Darling Astronomical Observatory (OAdM) is owned by the Catalan Government and operated by the Institute for Space Studies of Catalonia (IEEC). This article test rule out the null hypothesis that both samples (with 33 and is based on observations made in the Observatorios de Canarias del IAC with 36 values, respectively) are drawn from the same distribution. the 0.82 m IAC80 telescope operated on the island of Tenerife by the Instituto The p-values are higher than 0.3, far even from a lax threshold de Astrofísica de Canarias (IAC) in the Observatorio del Teide. This article is of statistical significance of 5%. Perhaps a larger sample could based on observations made with the SARA telescopes (Southeastern Associa- tion for Research in Astronomy), whose nodes are located at the Observatorios lead to a firmer conclusion in the future. de Canarias del IAC on the island of La Palma in the Observatorio del Roque This does not necessarily deny the hypothesis of Γ growth de los Muchachos; Kitt Peak, AZ under the auspices of the National Optical with period; instead it might indicate various levels of regolith Astronomy Observatory (NOAO); and Cerro Tololo Inter-American Observatory development on the surface, connected to age and/or collisional (CTIO) in La Serena, Chile. This project uses data from the SuperWASP archive. history, and space weathering. This could be further tested if we The WASP project is currently funded and operated by Warwick University and Keele University, and was originally set up by Queen’s University Belfast, the had sufficiently large samples of objects belonging to young and Universities of Keele, St. Andrews, and Leicester, the Open University, the Isaac old collisional families, since younger surfaces are expected to Newton Group, the Instituto de Astrofisica de Canarias, the South African Astro- have less developed regoliths (e.g. Delbo’ et al. 2014). The inter- nomical Observatory, and by STFC. Funding for the Kepler and K2 missions is pretation of our results might also be complicated by the temper- provided by the NASA Science Mission Directorate. The data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). ature dependence of the thermal conductivity, which we account STScI is operated by the Association of Universities for Research in Astronomy, for by assuming the scaling relation between Γ and heliocentric Inc., under NASA contract NAS5- 26555. Support for MAST for non-HST data distance given for example in Eq. (13) of Delbo’ et al.(2015) to is provided by the NASA Office of Space Science via grant NNX09AF08G and normalise the values of Γ to 1 au (these values are given in the by other grants and contracts. This publication makes use of data products from last column of Table2). This scaling is related to the T 3 depen- the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute dence of the thermal conductivity (e.g. Vasavada et al. 1999), and of Technology, funded by the National Aeronautics and Space Administration. 3/2 translates into a thermal inertia dependence of Γ ∝ T . How- The research leading to these results has received funding from the LP2012-31 ever, we note that Centaurs and TNOs (trans-Neptunian objects) and LP2018-7/2018 Lendület grants of the Hungarian Academy of Sciences. follow a different behaviour (Γ ∝ T 2; Lellouch et al. 2013), and Rozitis & et al.(2018) have found more extreme dependencies References in two near-Earth asteroids (Γ ∝ T 4.4 and Γ ∝ T 2.92), although Alí-Lagoa, V., Lionni, L., Delbo, M., et al. 2014, A&A, 561, A45 the fitted exponents have large error bars. More work is certainly Alí-Lagoa, V., Müller, T. G., Usui, F., & Hasegawa, S. 2018, A&A, 612, A85 needed in this direction. Binzel, R. P. 1987, Icarus, 72, 135 Our large observing campaign, targeted at about 100 aster- Blanco, C., Cigna, M., & Riccioli, D. 2000, Planet. Space Sci., 48, 973 oids with slow rotation and small amplitudes, already resulted Bottke, W. F., Durda, D. D., Nesvorný, D., et al. 2005, Icarus, 175, 111 Buchheim, R. K. 2007, Minor Planet Bull., 34, 68 in the gathering of around 10 000 h of photometric data, where Bus, S. J., & Binzel, R. P. 2002a, Icarus, 158, 146 hundreds of hours are needed for a unique lightcurve inversion Bus, S. J., & Binzel, R. P. 2002b, Icarus, 158, 106 solution for each of such targets. There are a few factors further Carry, B. 2012, Planet. Space Sci., 73, 98 limiting the number of targets with all the parameters uniquely Cibulková, H., Durech,ˇ J., Vokrouhlický, D., Kaasalainen, M., & Oszkiewicz, determined, like shape model imperfections, or thermal data D. A. 2016, A&A, 596, A57 Cutri, R. M., et al. 2012, VizieR Online Data Catalog: II/311 limited geometries. In the future we are planning to add more Delbo’, M., & Harris, A. W. 2002, Meteorit. Planet. Sci., 37, 1929 photometric data to our datasets accumulated already including Delbo’, M., & Tanga, P. 2009, Planet. Space Sci., 57, 259 photometry from the TESS (Transiting Exoplanet Survey Satel- Delbo’, M., Libourel, G., Wilkerson, J., et al. 2014, Nature, 508, 233 lite) mission, which should improve the uniqueness of many Delbo’, M., Mueller, M., Emery, J. P., Rozitis, B., & Capria, M. T. 2015, Asteroids IV, eds. P. Michel, F. E. DeMeo, & W. F. Bottke (Tucson, AZ: spin and shape solutions for targets not presented here. The University of Arizona Press), 107 TESS mission is observing in a similar cadence to the Kepler Dunham, D. W., Herald, D., Frappa, E., et al. 2016, NASA Planetary Data Space Telescope. 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A139, page 13 of 40 A&A 625, A139 (2019)

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

Table A.1. Observation details.

Date λ Phase Duration σ Observer Site angle (◦)(◦) (h) (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 h total

Notes. Columns are: mid-time observing date, ecliptic 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 is available at the CDS.

A139, page 14 of 40 A. Marciniak et al.: Thermal properties of slowly rotating asteroids

100 Hekate 100 Hekate P=27.069 h P=27.068 h 8,2 8,35 2018 Feb 1.1 Kepler Feb 2.1 Kepler Feb 3.3 Kepler 8,25 8,4 Feb 4.0 Kepler Feb 5.0 Kepler Feb 5.9 Kepler 8,3 May 19.1 Kepler 8,45 May19.8 Kepler May 20.8 Kepler May 21.8 Kepler 8,35 May 22.5 Kepler 8,5 8,4 2006 Nov 27.2 SuperWASP Relative C magnitude Nov 29.2 SuperWASP Relative R magnitude 8,55 8,45 Nov 30.2 SuperWASP Dec 14.2 SuperWASP Dec 15.2 SuperWASP Dec 16.2 SuperWASP 8,6 8,5 Dec 23.1 SuperWASP Dec 30.0 SuperWASP Zero time at: 2006 Nov 27.0467 UTC, LT corr. Zero time at: 2018 Jan 31.7150 UTC, LT corr. 8,55 8,65 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.1. Composite lightcurve of (100) Hekate from the year 2006. Fig. A.4. Composite lightcurve of (100) Hekate based on Kepler K2 data from the year 2018.

100 Hekate P=27.080 h 109 Felicitas -3,3 P = 13.191 h

9,4 -3,25

-3,2

2015 9,5 Jul 11.0 Bardon -3,15 Jul 12.1 Bardon Jul 13.1 Bardon Jul 14.1 Bardon Jul 15.1 Bardon Jul 16.0 Suhora Relative R magnitude -3,1 Jul 16.0 Terre. Jul 16.1 Bardon

Relative C magnitude 9,6 Jul 21.0 Suhora Aug 7.1 Bardon -3,05 Sep 1.9 Suhora 2006 Sep 14.9 OAdM Jul 1.1 SuperWASP Sep 19.9 OAdM Aug 30.8 SuperWASP Sep 20.9 OAdM Zero time at: 2015 Jul 11.9942 UTC, LT corr. Sep 14.9 SuperWASP -3 9,7 Zero time at 2006 Aug 30.7462 UTC, LT corr. Sep 20.8 SuperWASP 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Fig. A.2. Composite lightcurve of (100) Hekate from the year 2015. Fig. A.5. Composite lightcurve of (109) Felicitas from the year 2006.

100 Hekate P=27.072 h 109 Felicitas 7,75 P = 13.194 h -2,1

7,8 -2,05

7,85 -2

2016/2017 7,9 -1,95 Oct 3.1 Bardon Oct 7.0 Bardon Oct 8.0 Bardon 2015 Oct 28.1 Bardon -1,9 Oct 26.9 Bor.

Relative R magnitude 7,95 Oct 30.2 Bardon Oct 31.7 Bor. Oct 31.0 Bardon Nov 22.8 Bor. Nov 2.3 Tempe Dec 20.8 Bor. Nov 5.4 Tempe Relative R and C magnitude -1,85 Dec 21.8 OAdM Nov 6.3 Tempe 8 Dec 22.8 OAdM Nov 7.3 Tempe Dec 23.8 OAdM Nov 8.3 Tempe -1,8 Dec 24.8 OAdM Zero time at: 2016 Nov 2.1408 UTC, LT corr. Nov 9.3 Tempe Dec 25.8 OAdM Dec 4.2 Tempe Dec 26.8 OAdM 8,05 Zero time at 2015 Oct 31.6875, LT corr. Dec 27.8 OAdM 0 0,1 0,2 0,3 0,4 0,5 0,6 Dec0,7 5.2 Tempe 0,8 0,9 1 Jan 10.8 Bor. -1,75 Phase 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Fig. A.3. Composite lightcurve of (100) Hekate from the years 2016– 2017. Fig. A.6. Composite lightcurve of (109) Felicitas from the year 2015.

A139, page 15 of 40 A&A 625, A139 (2019)

109 Felicitas 195 Eurykleia

P = 13.188 h P=16.524 h 2008/2009 -4,5 2017 28 Oct 2008 Jan 22.3 CTIO 30 Oct 2008 Jan 27.1 Bor. -3,9 15 Nov 2008 Jan 29.1 Bor. 23 Nov 2008 Feb 16.1 Bor. 24 Nov 2008 Feb 22.4 Tempe 17 Dec 2008 17 Jan 2009 -4,4 Feb 23.4 Tempe 19 Jan 2009 Feb 24.4 Tempe Feb 25.4 Tempe -3,8 Feb 27.4 Tempe Mar 1.4 Tempe

-4,3 -3,7 Relative C magnitude Relative C and R magnitude

-4,2 -3,6 Zero time at 2017 Jan 28.9650 UTC, LT corr. 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 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 2017. Fig. A.10. Composite lightcurve of (195) Eurykleia from the year 2008.

109 Felicitas 195 Eurykleia P = 13.198 h -5 P=16.518 h 2013 -4,5 29 Sep, Bor. 30 Sep, Bor. 13 Oct, Bor. 3 Nov, Bor. 2 Dec, Bor. 17 Dec, Bor. -4,9 -4,4

Mar 16.5 Winer Mar 17.5 Winer 2018 May 4.3 Lowell May 5.3 Lowell -4,3 May 6.3 Lowell -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 magnitude May 19.3 Winer May 20.3 Lowell May 21.3 Winer -4,2 May 24.1 Winer May 25.3 Lowell -4,7 May 26.2 Lowell May 31.2 Lowell Jun 3.2 Lowell Zero time at 2018 May 21.1700 UTC, LT corr. Jun 13.2 Lowell Zero time at: 2013 Sep 29.9142 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.8. Composite lightcurve of (109) Felicitas from the year 2018. Fig. A.11. Composite lightcurve of (195) Eurykleia from the year 2013.

195 Eurykleia 195 Eurykleia P=16.522 h P=16.527 h 2015

-3,9 Sep 24.1 Bor. Nov 4.1 Bor. -3,4 Nov 5.1 Bor. Jan 5.0 Bor. Feb 14.9 Bor. Feb 17.9 Bor. Feb 18.0 Der. Mar 10.0 Bor. -3,8 -3,3

2004/2005 Nov 24.1 Sozzago Dec 4.1 Engar. Feb 1.9 Linhac. -3,7 Feb 2.8 Sozzago Feb 2.9 Linhac. Feb 3.9 Linhac. -3,2 Feb 14.9 Linhac. Feb 17.8 Sozzago

Relative R and C magnitude Feb 23.8 Hauts P. Feb 25.0 Blauvac Feb 25.9 Blauvac Relative C and R magnitude Feb 27.9 Blauvac Feb 28.9 Blauvac Mar 1.9 Hauts P. -3,6 Mar 8.9 Sozzago Mar 9.8 Tradate Mar 9.9 Sozzago -3,1 Zero time at: 2008 Oct 28.1912 UTC, LT corr. Mar 15.9 Linhac. Zero time at: 2015 Feb 14.7050 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 Phase 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Fig. A.9. Composite lightcurve of (195) Eurykleia from the years 2004– 2005. Fig. A.12. Composite lightcurve of (195) Eurykleia from the year 2015.

A139, page 16 of 40 A. Marciniak et al.: Thermal properties of slowly rotating asteroids

195 Eurykleia 301 Bavaria P=12.240 h 2015 P=16.52 h 2016 -2,7 May 24.4 Winer Apr 27.1 JKT Jun 4.4 Winer May 11.1 IAC80 -2,1 Jun 18.4 Winer May 25.0 IAC80 Sep 5.9 OAdM Jun 8.2 Winer Sep 6.8 Adi Jun 8.3 Winer Jun 18.2 Winer Sep 6.9 OAdM -2,6 Sep 10.9 OAdM Oct 2.8 OAdM Oct 3.8 OAdM -2

-2,5

-1,9 Relative R magnitude Relative C and R magnitude -2,4

-1,8 Zero time at: 2015 Sep 6.6933 UTC, LT corr. Zero time at: 2016 Apr 26.9767 UTC, LT corr. -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 Phase Phase

Fig. A.13. Composite lightcurve of (195) Eurykleia from the year 2016. Fig. A.16. Composite lightcurve of (301) Bavaria from the year 2015.

195 Eurykleia 301 Bavaria P=12.244 h P=16.52 h -3,1 -4,2 2016/2017 Oct 12.3 CTIO Oct 30.3 CTIO Nov 22.0 Bor Jan 18.1 CTIO -3 Feb 12.5 BOAO -4,1

-2,9 -4 Relative R magnitude Relative R and C magnitude -2,8 -3,9 2017 Aug 4.0 OAdM Aug 5.0 OAdM Aug 28.0 IAC80, Teide Zero time at: 2016 Nov 21.7929 UTC, LT corr. Zero time at: 2017 Aug 31.8633 UTC, LT corr. Aug 31.9 IAC80, Teide -2,7 -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.14. Composite lightcurve of (195) Eurykleia from the year 2017. Fig. A.17. Composite lightcurve of (301) Bavaria from the years 2016– 2017.

301 Bavaria 301 Bavaria P=12.243 h 2014 P=12.242 h -2,3 -3,1 24 Feb, Bor. 2018 2 Mar, Bor. 11 Mar, Bor. Jan 23.4 Winer 13 Mar, Bor. Mar 4.9 Bor. Mar 19.1 OASI 23 Mar, Organ M. Mar 20.1 OASI -2,2 1 Apr, Organ M. 22 May, Bor. -3

-2,1 -2,9 Relative C magnitude Relative R and C magnitude -2 -2,8

Zero time at: 2014 Feb 24.0821, LT corr. Zero time at: 2018 Jan 23.1642 UTC, LT corr. -1,9 -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 ar 2018. Fig. A.15. Composite lightcurve of (301) Bavaria from the year 2014. Fig. A.18. Composite lightcurve of (301) Bavaria from the year 2018.

A139, page 17 of 40 A&A 625, A139 (2019)

335 Roberta 380 Fiducia P=12.027 h 2013 P=13.71 h 9,5 29 Sep, Bor. 18 Oct, Organ M. 19 Oct, Bor. -3,3 19 Oct, Organ M. 25 Oct, Bor. 21 Nov, Organ M. 9,6 18 Dec, Organ M. 23 Dec, Bor.

9,7 -3,2 Relative C magnitude Relative C magnitude 9,8 2008 Feb 3.0 Villefagnan Feb 7.0 Villefagnan Feb 7.9 Villefagnan -3,1 Feb 9.0 Villefagnan 9,9 Feb 10.0 Villefagnan Zero time at: 2013 Sep 28.9158 UTC, LT corr. Zero time at: 2008 Feb 2.9179 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.19. Composite lightcurve of (335) Roberta from the year 2013. Fig. A.22. Composite lightcurve of (380) Fiducia from the year 2008.

335 Roberta 380 Fiducia P=12.027 h P=13.75 h -3,75 9,3

-3,7 9,4

-3,65

9,5

-3,6 2015 Nov 18.5 Winer Dec 9.0 Bor Feb 14.7 Bisei -3,55 Mar 9.9 Der Relative C magnitude 9,6 Mar 17.0 Der Relative C and R magnitude 2009 Apr 5.0 Blauvac -3,5 Apr 14.0 Blauvac 9,7 Zero time at: 2015 Feb 14.4592 UTC, LT corr. Zero time at: 2009 Apr 5.8204 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.20. Composite lightcurve of (335) Roberta from the year 2015. Fig. A.23. Composite lightcurve of (380) Fiducia from the year 2009.

335 Roberta 380 Fiducia P=12.024 h P=13.702 h 2014 -3,65 -2,8 2017 15 Jun, OAdM 19 Jun, OAdM Oct 16.0 Bor. 20 Jun, OAdM Oct 30.9 Bor. 21 Jun, OAdM -3,6 Nov 16.0 Der. 27 Jun, OAdM -2,7 29 Jun, OAdM 4 Jul, OAdM -3,55

-2,6 -3,5

-3,45 Relative R magnitude -2,5 Relative C and R magnitude

-3,4 -2,4 Zero time at: 2017 Oct 15.8788 UTC, LT corr. Zero time at: 2014 Jun 19.8678 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 lightcurve of (335) Roberta from the year 2017. Fig. A.24. Composite lightcurve of (380) Fiducia from the year 2014.

A139, page 18 of 40 A. Marciniak et al.: Thermal properties of slowly rotating asteroids

380 Fiducia 380 Fiducia P=13.716 h P=13.728 h -3,5 -3,8 2015/2016 Aug 31.1 Bor. Sep 1.1 Bor. Sep. 28.0 Bor. -3,4 Dec 30.8 Bor. -3,7 Jan 22.3 Bor. Feb 25.2 Winer

-3,3 -3,6

Relative C magnitude -3,2 -3,5 2018 Relative C and R magnitude Mar 4.1 Suhora Mar 26.1 Bor. Apr 10.0 Bor. Apr 13.0 Bor. -3,1 -3,4 Apr 15.0 Bor. Zero time at: 2015 Aug 31.0021 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.25. Composite lightcurve of (380) Fiducia from the years 2015– Fig. A.28. Composite lightcurve of (380) Fiducia from the year 2018. 2016 under large phase angle.

380 Fiducia 468 Lina P=13.719 h P = 16.48 h -3,1 10 2015 Oct 11.0 Bor. Oct 31.0 Bor. 10,05 -3

10,1

-2,9 10,15 Relative C magnitude -2,8 Relative C magnitude 10,2

2006 Jan 3.9 Hauts Patys 10,25 Jan 4.9 Hauts Patys -2,7 Jan 13.9 Hauts Patys Jan 14.9 Hauts Patys Zero time at: 2015 Oct 31.7696 UTC, LT corr. Zero time at: 2006 Jan 14.7246 UTC, LT corr. 10,3 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.26. Composite lightcurve of (380) Fiducia from the year 2015 Fig. A.29. Composite lightcurve of (468) Lina from the year 2006. under small phase angle.

380 Fiducia 468 Lina P=13.717 h P=16.48 h -3,4 8,25

8,3 -3,3

8,35

-3,2 8,4

2016/2017 Nov 29.1 Bor. -3,1 Dec 13.0 Bor. Relative C magnitude 8,45 2007 Dec 28.1 Bor. Relative C and R magnitude Jan 8.9 Bor. Feb 11.1 SM Jan 26.9 Bor. Feb 15.2 SM Feb 8.2 CTIO Feb 17.2 SM Feb 15.0 Bor. 8,5 Feb 20.1 SM -3 Mar 11.8 Bor. Feb 20.3 SM Mar 24.8 Bor. Feb 21.2 SM Zero time at: 2016 Dec 12.9042 UTC, LT corr. Zero time at: 2007 Feb 11.0392 UTC, LT corr. Feb 21.3 SM 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.27. Composite lightcurve of (380) Fiducia from the years 2016– Fig. A.30. Composite lightcurve of (468) Lina from the year 2007. 2017.

A139, page 19 of 40 A&A 625, A139 (2019)

468 Lina 538 Friederike P=16.56 h 2014 P=46.724 h -3,8 -4,1 24 Mar, OAdM Feb 9, OAdM 23 May, OAdM 2014 Feb 17, OAdM 26 May, OAdM Feb 21, OAdM Feb 22, OAdM 12 Jun, OAdM -3,75 Feb 23, OAdM Feb 27, OAdM -4 Mar 8, Bor. Mar 16, OAdM -3,7 Mar 17, OAdM Mar 23, OAdM Mar 27, OAdM Apr 17, OAdM -3,65 -3,9 Apr 18, OAdM Apr 23, OAdM Apr 25, OAdM

Relative R magnitude -3,6 Relative R and C magnitude -3,8

-3,55

Zero time at: 2014 May 23.8908 UTC, LT corr. Zero time at: 2014 Mar 17.8204 UTC, LT corr. -3,5 -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.31. Composite lightcurve of (468) Lina from the year 2014. Fig. A.34. Composite lightcurve of (538) Friederike from the year 2014.

468 Lina 538 Friederike P = 16.480 h -3,3 P=46.745 h -3,4

-3,25

-3,3 -3,2

2015 Mar 16.1 Bor -3,15 2015 Mar 17.0 Bor May 13.4 Winer -3,2 Apr 16.4 Winer May 17.5 Winer Apr 27.4 Winer May 26.4 Winer Apr 28.3 Winer -3,1 May 31.4 Winer May 7.3 Winer Jun 15.4 Winer May 8.3 Winer Relative L and R magnitude Jun 16.4 Winer May 15.8 Adi Relative R and C magnitude Jun 26.1 Teide -3,1 May 16.8 Adi Jul 1.1 Teide May 17.2 Winer -3,05 Jul 18.0 OAdM Jul 23.0 OAdM May 20.2 Winer Sep 9.9 OAdM May 21.2 Winer Zero time at: 2015 May 13.3612 UTC, LT corr. Sep 14.9 OAdM Zero time at: 2015 Apr 27.1171 UTC, LT corr. -3 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 -3 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase

Fig. A.32. Composite lightcurve of (468) Lina from the year 2015. Fig. A.35. Composite lightcurve of (538) Friederike from the year 2015.

468 Lina P = 16.485 h 538 Friederike -3,4 2017 P=46.73 h -3,4 Aug 30.0 Bor. Sep 1.1 OAdM -3,35 Sep 2.0 OAdM Sep 3.0 OAdM Sep 19.0 OAdM -3,3 Sep 20.0 OAdM -3,3 Sep 25.9 OAdM Sep 27.9 OAdM Oct 7.0 OAdM Oct 8.0 OAdM Oct 21.0 OAdM -3,25 2016/2017 Oct 25.9 OAdM -3,2 Nov 21.5 SOAO Sep 7.1 Bor. Dec 2.5 SOAO Nov 2.3 Tempe -3,2 Nov 5.3 Tempe Nov 6.4 Tempe

Relative R and C magnitude Nov 7.3 Tempe

Nov 8.3 Tempe Relative R and C magnitude Nov 9.3 Tempe -3,1 -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. -3,1 Zero time at: 2017 Aug 30.9121 UTC, LT corr. 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 -3 Phase 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Fig. A.33. Composite lightcurve of (468) Lina from the years 2016– 2017. Fig. A.36. Composite lightcurve of (538) Friederike from the year 2017.

A139, page 20 of 40 A. Marciniak et al.: Thermal properties of slowly rotating asteroids

538 Friederike 653 Berenike P=46.74 h P=12.488 h -3,5 -3,5

-3,4 -3,4

-3,3

2018 -3,3 Oct 18.1 Bor. Oct 19.0 Bor. Nov 1.1 Bor. -3,2 Nov 4.1 OAdM

Nov 6.1 Bor. Relative R magnitude Nov 7.1 Bor. 2015 Relative R and C magnitude -3,2 Nov 10.4 Organ M. Jun 18.0 OAdM Nov 17.4 Organ M. Jun 19.0 OAdM Nov 18.0 Bor. -3,1 Jun 21.0 OAdM Nov 29.3 Organ M. Jun 29.1 Teide Dec 5.9 Bor. Jul 5.0 Teide Dec 11.2 OAdM Aug 10.9 OAdM Zero time at: 2018 Oct 17.9492 UTC, LT corr. Dec 14.3 Organ M. Zero time at: 2015 Jun 17.9025 UTC, LT corr. Aug 11.9 OAdM -3,1 -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.37. Composite lightcurve of (538) Friederike from the year 2018. Fig. A.40. Composite lightcurve of (653) Berenike from the year 2015.

653 Berenike 653 Berenike P=12.487 h P=12.487 h -4 8,5

-3,9 8,6

-3,8 8,7

-3,7 8,8 2016 Relative R magnitude

Relative C magnitude Aug 13.2 ORM 2005 Sep 6.0 OAdM Jul 3.1 Engar. Sep 7.0 OAdM -3,6 8,9 Jul 9.0 Engar. Sep 12.0 OAdM Aug 7.0 Engar. Oct 10.0 OAdM Aug 9.0 Engar. Oct 11.8 OAdM Oct 15.9 OAdM Zero time at: 2005 Aug 8.8438 UTC, LT corr. Zero time at: 2016 Aug 23.0958 UTC, LT corr. -3,5 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.38. Composite lightcurve of (653) Berenike from the year 2005. Fig. A.41. Composite lightcurve of (653) Berenike from the year 2016.

653 Berenike 653 Berenike P=12.485 h P=12.481 h -4,5 -3,4

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

-4,2 -3,1 Relative C magnitude

Relative C magnitude 2017/2018 Oct 31.2 CTIO -4,1 Nov 18.1 Bor Jan 15.8 Bor. -3 Jan 15.6 SOAO Feb 26.8 Bor. Zero time at: 2018 Feb 26.7421 UTC, LT corr. Zero time at: 2014 Mar 26.8125 UTC, LT corr. -4 -2,9 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.42. Composite lightcurve of (653) Berenike from the years 2017– Fig. A.39. Composite lightcurve of (653) Berenike from the year 2014. 2018.

A139, page 21 of 40 A&A 625, A139 (2019)

673 Edda 673 Edda P=22.32 h P=22.340 h 10,9 -3,95 2016 Jun 8.4 Winer Jun 19.4 Lowell 10,95 -3,9 Jul 10.0 OAdM Jul 11.0 OAdM Jul 13.0 OAdM Jul 15.0 OAdM Jul 16.0 OAdM 11 -3,85 Jul 17.0 OAdM Jul 28.1 Teide Aug 3.0 OAdM Sep 22.9 ORM 11,05 -3,8 Relative R magnitude 11,1 Relative R magnitude -3,75

2005 Mar 10.9 Engar. 11,15 -3,7 Mar 12.0 Engar. Mar 12.9 Engar. Zero time at: 2005 Mar 10.7754 UTC, LT corr. Zero time at: 2016 Sep 22.8104 UTC, LT corr. 11,2 -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.43. Composite lightcurve of (673) Edda from the year 2005. Fig. A.46. Composite lightcurve of (673) Edda from the year 2016.

673 Edda 673 Edda P=22.34 h P=22.338 h 10,05 9,75 2017 Sep 30.1 Bor. Oct 2.3 Kitt Peak 10,1 9,8 Oct 9.9 Bor. Oct 17.0 Der. Nov 23.9 Bor.

10,15 9,85

10,2 9,9

Relative R magnitude 10,25 9,95 Relative C and R magnitude

2007 10,3 Sep 16.9 Blauvac 10 Sep 20.0 Blauvac Zero time at: 2007 Sep 16.8104 UTC, LT corr. Zero time at: 2017 Oct 16.8379 UTC, LT corr. 10,35 10,05 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.44. Composite lightcurve of (673) Edda from the year 2007. Fig. A.47. Composite lightcurve of (673) Edda from the year 2017.

673 Edda 834 Burnhamia P=22.338 h P = 13.89 h -3,05 8,9 2013/2014 Nov 25.1 Bor. Jan 23.8 Bor. Feb 2.9 Bor. -3 Feb 22.8 Bor. Feb 23.0 Bor. Feb 25.8 Bor. 9 -2,95

-2,9 9,1 Relative C magnitude -2,85 Relative R magnitude

9,2 2005 May 5.0 Engar. -2,8 May 6.0 Engar. May 8.0 Engar. Zero time at: 2013 Nov 25.0275 UTC, LT corr. Zero time at: 2005 May 5.8846 UTC, LT corr. -2,75 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.45. Composite lightcurve of (673) Edda from the years 2013– Fig. A.48. Composite lightcurve of (834) Burnhamia from the year 2014. 2005.

A139, page 22 of 40 A. Marciniak et al.: Thermal properties of slowly rotating asteroids

834 Burnhamia 834 Burnhamia P = 13.88 h P = 13.879 h 9,4

-4

9,5

-3,9

9,6

-3,8

Relative R magnitude 2016

2006 Relative R and C magnitude Jan 26.4 Winer 9,7 Jan 27.5 Winer Oct 8.0 Engar. Feb 18.4 Winer Oct 9.9 Engar. Feb 23.5 Winer Oct 10.9 Engar. -3,7 Mar 7.5 Winer Mar 17.1 Bor. Zero time at: 2006 Oct 7.8088 UTC, LT corr. Zero time at: 2016 Jan 26.3400 UTC, LT corr. May 7.9 Bor. 9,8 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.49. Composite lightcurve of (834) Burnhamia from the year Fig. A.52. Composite lightcurve of (834) Burnhamia from the year 2006. 2016.

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

-2,9 -4,2

-2,8 -4,1

Relative R magnitude 2017

Relative C magnitude Jul 29.0 OAdM -2,7 Jul 30.0 OAdM Jul 31.0 Suhora -4 Aug 1.0 Suhora 2013/2014 Aug 2.0 OAdM Dec 3.1 Bor. Aug 3.0 OAdM Jan 8.2 Winer Zero time at: 2017 Jul 31.8262 UTC, LT corr. Zero time at: 2013 Dec 3.0000 UTC, LT corr. Mar 24.2 Winer -2,6 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 -3,9 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig. A.53. Composite lightcurve of (834) Burnhamia from the year Fig. A.50. Composite lightcurve of (834) Burnhamia from the years 2017. 2013–2014.

834 Burnhamia P = 13.876 h -5,7 2014/2015 Nov 25.4 Winer Nov 26.4 Winer Dec 17.1 OAdM Dec 20.1 OAdM 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

-5,5 Relative R magnitude

-5,4

Zero time at: 2014 Nov 25.3242 UTC, LT corr. -5,3 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase

Fig. A.51. Composite lightcurve of (834) Burnhamia from the years 2014–2015.

A139, page 23 of 40 A&A 625, A139 (2019)

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

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 All data 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

A139, page 24 of 40 A. Marciniak et al.: Thermal properties of slowly rotating asteroids

Table B.3. Summary of TPM results for (195) Eurykleia fitting different subsets of data.

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 Only W4 0.23 81 5 Low (0.29) Artificially low 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.

Notes. For example, the “Excluding W4” label refers to the fact that the W4 data were not used to optimise the χ2 in that case.

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

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.

A139, page 25 of 40 A&A 625, A139 (2019)

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 Bad fit +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 Bad fit +5 +25 ∼ AM 2 WISE only 0.72 76−2 20−20 Extr. high 1 rms >0.45

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

A139, page 26 of 40 A. 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) 7 8

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 lightcurves Fig. B.4. Asteroid (335) Roberta’s WISE data and model thermal for shape model 1. Data error bars are 1σ. Table B.1 summarises the lightcurves for shape model 1 and the corresponding best-fitting solu- TPM analysis. tions (very low thermal inertia of 15 SIu).

9.5 9 W4 data W4 data Best fit AM 1 9 Best fit AM 2 (all data used) 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) Eurykleia’s Fig. B.5. Asteroid (380) Fiducia’s WISE data and model thermal shape model AM 2. The different models resulted from fitting different lightcurves for shape model 1 and the corresponding best-fitting solu- subsets of data. Table B.3 contains the corresponding thermo-physical tions (very low thermal inertia of 15 SIu; see Table B.6). parameters.

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

3.8 6

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 and lightcurves for shape model 2’s corresponding best-fitting solutions (see W4 data (see Table B.4). Table B.7).

A139, page 27 of 40 A&A 625, A139 (2019)

0.75 W3 data W3 data Best fit AM 2 1.4 Best fit AM 2 0.7

1.2 0.65

0.6 1 Flux (Jy) Flux (Jy)

0.55 0.8

0.5 0.6

0.45 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 Rotational phase Rotational phase

1.7 W4 data W4 data 3.4 Best fit AM 2 Best fit AM 2 1.6 3.2

1.5 3

1.4 2.8 Flux (Jy)

Flux (Jy) 1.3 2.6

1.2 2.4

1.1 2.2

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

Fig. B.7. Asteroid (673) Edda’s WISE data and model thermal Fig. B.8. Asteroid (834) Burnhamia’s WISE data and model thermal lightcurves for shape model 2’s best-fitting solution (very low thermal lightcurves for shape model 2’s best-fitting solution (see Table B.11). inertia of 3 SI units).

A139, page 28 of 40 A. 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: observation 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 of surface roughness.

A139, page 29 of 40 A&A 625, A139 (2019)

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: observation to model ratios versus wavelength, heliocentric distance, rotational phase, and phase angle for model AM 1 (AM 2 was rejected). Right: χ2 versus thermal inertia curves for model AM1.

A139, page 30 of 40 A. 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 “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.

A139, page 31 of 40 A&A 625, A139 (2019)

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.

A139, page 32 of 40 A. 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: observation to model ratios versus wavelength, heliocentric distance, 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).

A139, page 33 of 40 A&A 625, A139 (2019)

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 heliocentric 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).

A139, page 34 of 40 A. 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 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).

A139, page 35 of 40 A&A 625, A139 (2019)

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 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).

A139, page 36 of 40 A. 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: observation to model ratios versus wavelength, heliocentric distance, 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.

A139, page 37 of 40 A&A 625, A139 (2019)

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.

A139, page 38 of 40 A. 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 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).

A139, page 39 of 40 A&A 625, A139 (2019)

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).

A139, page 40 of 40