(25143) Itokawa: The Power of Radiometric Techniques for the Interpretation of Remote Thermal Observations in the Light of the Hayabusa Rendezvous Results
((25143) Itokawa: The Power of Radiometric Techniquesfor the Interpretation of Remote Thermal Observationsin the Light of the Hayabusa Rendezvous Results ∗ Thomas G.
M¨uller
Max-Planck-Institut f¨ur extraterrestrische Physik, Giessenbachstraße, 85748 Garching, [email protected]
Sunao
Hasegawa
Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai,Chuo-ku, Sagamihara [email protected] andFumihiko
Usui
Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo,Bunkyo-ku, Tokyo [email protected] (Received ; accepted )
Abstract
The near-Earth asteroid (25143) Itokawa was characterised in great detail by theJapanese Hayabusa mission. We revisited the available thermal observations in thelight of the true asteroid properties with the goal to evaluate the possibilities andlimitations of thermal model techniques. In total, we used 25 published ground-basedmid-infrared photometric observations and 5 so far unpublished measurements fromthe Japanese infrared astronomical satellite AKARI in combination with improvedH-G values (absolute magnitude and slope parameter). Our thermophysical model(TPM) approach allowed us to determine correctly the sense of rotation, to estimatethe thermal inertia and to derive robust effective size and albedo values by only usinga simple spherical shape model. A more complex shape model, derived from light-curve inversion techniques, improved the quality of the predictions considerably andmade the interpretation of thermal light-curve possible. The radiometrically derivedeffective diameter value agrees within 2% of the true Itokawa size value. The combi-nation of our TPM and the final (25143) Itokawa in-situ shape model was then usedas a benchmark for deriving and testing radiometric solutions. The consolidated valuefor the surface-averaged thermal inertia is Γ = 700 ±
200 J m − s − . K − . We found1 a r X i v : . [ a s t r o - ph . E P ] A p r hat even the high resolution shape models still require additional small-scale rough-ness in order to explain the disk-integrated infrared measurements. Our descriptionof the thermal effects as a function of wavelengths, phase angle, and rotational phasefacilitates the planning of crucial thermal observations for sophisticated characteri-zation of small bodies, including other potentially hazardous asteroids. Our analysisshows the power of radiometric techniques to derive the size, albedo, thermal inertia,and also spin-axis orientation from small sets of measurements at thermal infraredwavelengths. Key words: infrared: solar system — minor planets, asteroids: individual:(25143) Itokawa — radiation mechanisms: thermal — techniques: photometric
1. Introduction
The Near-Earth asteroid (NEA) (25143) Itokawa (1998 SF36) is one of the best studiedasteroids in our Solar System. It was the sample return target of the Japanese Hayabusa(MUSES-C) mission. The spacecraft was in close proximity to the asteroid from Septemberthrough early December 2005. As a result of the encounter, the asteroid has been characterisedin great detail: (25143) Itokawa is an irregularly formed body consisting of a loose pile of rubblerather than a solid monolithic asteroid (Fujiwara et al. 2006; Saito et al. 2006; Abe et al. 2006b).Its appearance is boomerang-shaped and composed of two distinct parts with faceted regionsand a concave ring structure in-between (Demura et al. 2006). Recently, the detection of YORPspin-up revealed that the two distinct parts of Itokawa have different densities and are likely tobe two merged asteroids (Lowry et al. 2014). Its effective diameter (of an equal volume sphere)is 327.5 ± ± × m ; Fujiwara et al. 2006). This compares verywell with the pre-encounter size prediction obtained via radiometric techniques by M¨uller etal. (2005) (M05 hereafter) of D eff =320 ±
30 m. The radar size prediction is about 16% too high(Ostro et al. 2004; Ostro et al. 2005). The mass is estimated as (3.58 ± × kg, implyinga bulk density of (1.95 ± − (Abe et al. 2006b). The retrograde pole orientation inecliptic coordinates is ( λ pole , β pole ) = (128.5 ◦ , -89.66 ◦ ), with a 3.9 ◦ margin of error (Demura etal. 2006). Itokawa is classified as an S IV-type asteroid via ground-based near-infrared (NIR)spectroscopy (Binzel et al. 2001), a type common in the inner portion of the asteroid belt. Thesemeasurements at mineralogically diagnostic wavelength show similarities to ordinary chondritesand/or primitive achondrite meteorites. Hayabusa confirmed the S-class asteroid characteristicsand revealed an olivine rich mineral assemblage of the surface, similar to LL5 or LL6 chondrites(Abe et al. 2006a; Okada et al. 2006). The most recent geometric albedo estimate of p V =0.29 ± ∗ Based on observations with AKARI, a JAXA project with the participation of ESA. ∼ − s − . K − ) and powdery surface like lunar regolith (Γ ∼
40 J m − s − . K − )(Yano et al. 2006; Noguchi et al. 2010). M05 used a sample of remote, disk-integrated thermalmeasurements to derive the average thermal inertia of Itokawa’s top surface layer. They founda thermal inertia value of roughly 750 J m − s − . K − . A study by Mueller (2007) found a verysimilar value of 700 J m − s − . K − . Gundlach & Blum (2013) combined the thermal inertiavalue with Itokawa’s known size and low gravitational acceleration on the surface to determinea mean surface particle radius of 21 +3 − mm which is in nice agreement with in-situ observationspresented by Yano et al. (2006) and Kitazato et al. (2008). One aspect of our work here wasto test the derived (pre-Hayabusa) thermal properties (mainly the object’s thermal inertia) inthe light of the in-situ results.We re-visit the available remote, disk-integrated thermal data, comprising ground-basedobservations in standard N- and Q-band filters with IRTF/MIRSI and ESO/TIMMI2, andfrom AKARI, the Japanese infrared astronomical satellite (Murakami et al. 2007). The goalof our study is to describe the possibilities and limitations of radiometric methods when usingremote, disk-integrated data, as available or easily obtainable for most of the minor bodies. Thederived values are then directly compared to the in-situ results from the Hayabusa-mission. Bycomparing the results from the radiometric techniques with the in-situ results, we validate modeltechniques and provide observing strategies for future applications to other targets, includingpotentially hazardous asteroids (PHAs).Section 2 gives an overview of the existing thermal observations and describes the so farunpublished observations by AKARI. In Section 3 we describe the different TPM applicationsand optimisation processes which we applied to the full dataset of remote thermal observationsand present the derived results: Section 3.1 describes briefly the thermophysical model (TPM)and the range of possible input parameters. In a first analysis step we use a spherical shapemodel with a range of pro- and retrograde spin-vectors (Section 3.2). In the second step(Section 3.3) we added the shape and spin-vector information derived from light-curve inversiontechniques. In the third step (Section 3.4) we used the true shape model and spin-vector solutionas provided by the Hayabusa mission. The flux predictions from the best TPM solution arethen compared with the available observations. In Section 4 we inter-compare different shapesolutions with respect to thermal light-curves and predict the behaviour of the spectral energydistribution (SED), the thermal light-curve amplitude, shape, and the thermal beaming effect.In Section 5 we discuss the potential and the limitation of the radiometric methods, using thevarious levels of information. The summary and transfer of applications to other targets isgiven in Section 6. 3 able 1. Summary of mid-IR observing sets for asteroid (25143) Itokawa.
Time † Filter r ‡ ∆ § α (cid:107) No ∗ [UT] Band [AU] [AU] [ ◦ ] Remarks21 2004/Jul/10 11:45 N11.7 1.060399 0.049983 +28.32 IRTF/MIRSI (Mueller M. et al. in press)
22 2004/Jul/10 11:48 N11.7 1.060407 0.049990 +28.31 IRTF/MIRSI (Mueller M. et al. in press)
23 2004/Jul/10 13:32 N11.7 1.060673 0.050243 +28.18 IRTF/MIRSI (Mueller M. et al. in press)
24 2004/Jul/10 13:41 N9.8 1.060696 0.050266 +28.17 IRTF/MIRSI (Mueller M. et al. in press)
25 2004/Jul/10 13:51 N9.8 1.060721 0.050290 +28.16 IRTF/MIRSI (Mueller M. et al. in press)
26 2007/Jul/26 11:29 N4 1.053777 0.281244 -73.49 AKARI (this work) ∗∗
27 2007/Jul/26 11:29 S7 1.053777 0.281244 -73.49 AKARI (this work) ∗∗
28 2007/Jul/26 11:28 S11 1.053774 0.281244 -73.49 AKARI (this work) ∗∗
29 2007/Jul/26 13:09 L18 1.054027 0.281282 -73.43 AKARI (this work) ∗∗
30 2007/Jul/26 13:12 L24 1.054035 0.281283 -73.43 AKARI (this work) ∗∗ Notes. ∗ Observations with running numbers 1-20 are listed in table 1 in M05. † The times are mid observing times in theobserver’s time frame. ‡ The heliocentric distance. § The observer-centric distance. (cid:107)
The phase angles, negative beforeopposition and positive after.
The observations in Mueller M. et al. (in press) have been shifted to the observer’s timeframe by adding 25 s to the light-time corrected times given in the publication. ∗∗ The geometry is given by the geocentriccalculation.
2. Thermal Observations and Input Data
We combine five previously published mid-infrared observations by Mueller M. et al.(in press) with 20 observations by M05 and five dedicated AKARI observations. The M05 data(table 1 & 3 in M05) have running indices from 1 to 20. The additional data presented hereare labeled 21-25 and 26-30 respectively (table 1 and table 2).
Mueller M. et al. (in press) presented a set of five N-band observations which we in-cluded in our calculations. For the entries in table 1 and table 2 (numbers 21-25) we usedthe monochromatic, colour-corrected fluxes (but now in Jansky-units) and calculated the trueobserving times (Mueller M. et al. in press gave times which were corrected for 1-way light-time,i.e., in the asteroid time frame). Mueller (2007) mentioned that they had taken the observationsat a relatively high level of atmospheric humidity. In addition, all observations were taken atair-masses larger than 2 due to technical problems at meridian transit.
Asteroid (25143) Itokawa was observed on July 27, 2007 by the NIR, MIR-S, and MIR-L channels on the infrared camera IRC (Onaka et al. 2007) on-board AKARI. During one4ointed observation, all three IRC channels obtained images simultaneously, covering differentwavelength ranges. The NIR and MIR-S channels share the same field of view, while theMIR-L channel observes a region which is ∼ (cid:48) away from the field centre of the NIR andMIR-S channels. In total, two pointed observations on Itokawa were carried out to obtain datain all three channels. The Astronomical Observation Template (AOT) IRC02 for dual-filterphotometry (see Onaka et al. 2007 for details) was used. As a result, observations for Itokawawith the NIR, MIR-S, and MIR-L were performed in the filters N3 (reference wavelength of3.2 µ m, but not used here for our thermal analysis), N4 (4.1 µ m), S7 (7.0 µ m), S11 (11.0 µ m),L15 (15.0 µ m) and L24 (24.0 µ m) with effective bandwidths of 0.9, 1.5, 1.8, 4.1, 6.0 and 5.3 µ m,respectively. The projected area of the NIR channels was about 10.0 (cid:48) × (cid:48) which correspondsto an angular resolution of 1.5 (cid:48)(cid:48) /pixel. The MIR-S channels of the IRC have pixel sizes ofabout 2.3 (cid:48)(cid:48) /pixel, giving a field of view about 10.0 (cid:48) × (cid:48) . The MIR-L channel was usedwith a image scale of 2.4 (cid:48)(cid:48) /pixel, giving a 10.2 (cid:48) × (cid:48) sky field. For the data processing theIRC imaging data pipeline was used. The AKARI telescope was not able to track movingobjects such as comets and asteroids. Therefore, a centroid determination in combination witha standard shift-and-add technique was performed, followed by median processing to obtainbetter photometric accuracy. Aperture photometry on IRC images was carried out using theAPPHOT task of IRAF thorough circular aperture radii of 10.0 (in the NIR channel) and7.5 (in the MIR-S and MIR-L channels) pixels, which are also used for the standard starflux calibration. The resulting astronomical data units were converted to the calibrated fluxdensities by using the IRC flux calibration constants in the Revisions of the IRC conversionfactors . Colour differences between calibration stars and Itokawa were not negligible due tothe wide bandwidths of the IRC. Colour correction factors were obtained using both predictedthermal flux of Itokawa and the relative spectral response functions for IRC. Colour correctionfluxes of Itokawa were obtained by dividing the quoted fluxes by 1.453 in N4 band, 1.020 inS7 band, 0.956 in S11 band, 0.960 in L18 band, and 1.079 in L24 band. The observationalresults are summarised in table 2 (numbers 26-30) and further details about AKARI asteroidobservations and catalogued data are given in Usui et al. (2011); Hasegawa et al. (2013).
3. Thermophysical Modelling
We applied the radiometric technique as described in M05. Via a χ -process, with di-ameter and thermal inertia as free parameters , we searched for the best solution to match AKARI IRC Data Users Manual ver.1.3, We also solve for the geometric albedo, but is not considered as a free parameter since it is tightly connectedto the H-magnitude via the size information: p V = 10 (2 · log ( S ) − · log ( D eff ) − . · H V ) , with the Solar constant able 2. Summary of the available thermal infrared observations of asteroid (25143) Itokawa. λ c FD σ err No ∗ Filter [ µ m] [Jy] [Jy] Remarks21 N11.7 11.7 0.762 0.100 IRTF/MIRSI (Mueller M. et al. in press) †
22 N11.7 11.7 0.721 0.091 IRTF/MIRSI (Mueller M. et al. in press) †
23 N11.7 11.7 0.913 0.114 IRTF/MIRSI (Mueller M. et al. in press) †
24 N9.8 9.8 0.791 0.125 IRTF/MIRSI (Mueller M. et al. in press) †
25 N9.8 9.8 0.570 0.122 IRTF/MIRSI (Mueller M. et al. in press) †
26 N4 4.1 0.00032 0.00025 AKARI (this work)27 S7 7.0 0.00469 0.00028 AKARI (this work)28 S11 11.0 0.01422 0.00053 AKARI (this work)29 L15 15.0 0.02137 0.00079 AKARI (this work)30 L24 24.0 0.01947 0.00120 AKARI (this work)
Notes. ∗ Observations with running numbers 1-20 are listed in table 1 in M05. † The fluxdensities in Mueller M. et al. (in press) have been converted to Jansky. all thermal observations listed in Sect. 2 simultaneously: χ = 1 / ( N − ν )Σ(( obs − mod ) ), with ν being the number of free parameters (here ν =2, with size and thermal inertia as free pa-rameters). The detailed steps are described in M¨uller et al. (2011). The TPM is detailed byLagerros (1996); Lagerros (1997); Lagerros (1998a); Lagerros (1998b); Harris & Lagerros (2002).It places the asteroid at the true illumination and observing geometry. For each surface elementthe solar insolation is taken into account and the amount of reflected light and thermal emissionare calculated, controlled by the albedo, the H-G values, the surface roughness (parameterisedby ρ , the r.m.s. of the surface slopes and f , the fraction of the surface covered by craters) andthe thermal inertia Γ. For the temperature calculation the one-dimensional vertical heat con-duction (controlled by the thermal inertia Γ) into the surface is taken into into account. Thetreatment of heat conduction inside the spherical section craters is approximated by using thebrightness temperature relations as a function of the thermal parameter Θ for a flat surface(Lagerros 1998a). In this way it is possible to separate the beaming from the heat conductionwhich is relevant for computation speed reasons. A summary of the influences of the thermalparameters as a function of wavelength and as a function of phase angle is given in M¨uller(2002). The technique to determine thermal properties from a set of thermal observations was S = 1366 W m − . The thermal inertia Γ is defined as √ κρc , where κ is the thermal conductivity, ρ is the density, and c is theheat capacity. The thermal parameter Θ is defined as (Γ √ ω ) / ( (cid:15)σT ss ), where Γ is the thermal inertia, ω is the angularvelocity of rotation and T ss is the sub-solar temperature. able 3. Summary of general TPM input parameters and applied variations.
Range Units/Remarks M05 valueΓ 0...2500 [J m − s − . K − ] 750thermal inertia ρ f (cid:15) λ -independent emissivity 0.9 H V +0 . − . [mag] 19.9Bernardi et al. (2009)G 0.21 +0 . − . Bernardi et al. (2009) 0.21P sid ± already successfully applied for large main-belt asteroids by e.g., Spencer et al. (1989); M¨uller& Lagerros (1998); M¨uller & Lagerros (2002); M¨uller et al. (1999); O’Rourke et al. (2012) anda range of near-Earth asteroids (e.g., M¨uller et al. 2011; M¨uller et al. 2012; M¨uller et al. 2013).The general TPM input parameters and parameter ranges are listed in table 3. Thefirst three parameters show the physically meaningful range for thermal properties (see e.g.,Lagerros 1998b). The constant emissivity is a standard value used in radiometric techniqueswhen applied to mid-infrared data (e.g., Lebofsky et al. 1986). The last three values are derivedfrom visual photometric measurements. The shapes and spin-axis orientations are not known for most of the asteroids. It istherefore very instructive to start the radiometric technique with the simplest shape modelto evaluate the possibilities and limitations of such a simple approach. In a first attempt tointerprete the thermal observations we use a spherical shape model with a range of pro- andretrograde spin-vector orientations ( β SVecl = ± ◦ , arbitrary λ SVecl ) and the values specifiedin table 3. Figure 1 shows the implementation of this model for the observation Nr. 12 (table 1in M05) at a phase angle of 54 ◦ (01-Jul-2004 06:03 UT) for a +90 ◦ prograde (top) and − ◦ retrograde (bottom) sense of rotation. The temperature pictures, as seen from the observer,are very different and consequently also the connected disk-integrated thermal fluxes. Theillustration shows that the combination of thermal observations from before and after opposition(with either warm or cold terminator) can indicate the true sense of rotation.The χ -figure (figure 2) shows that the best agreement between observations and model7 Temperature [K], spherical shape, prograde rotation y [km] z [ k m ] −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−0.2−0.15−0.1−0.0500.050.10.150.2 Temperature [K], spherical shape, retrograde rotation y [km] 160180200220240260280300320
Fig. 1.
The pro- and retrograde implementation of the spherical shape model for the epoch 01-Jul-200406:03 UT. For the temperature calculation a thermal inertia of 700 J m − s − . K − has been used.The viewing geometry is in the ecliptic coordinate system (spin vector perpendicular to the eclip-tic plane) as seen from Earth, projected on the sky. The Sun is at a phase angle of 54 ◦ . Theasteroid’s apparent position (Earth-centred) was 312 ◦ ecliptic longitude and -48 ◦ ecliptic latitude. Fig. 2.
The thermal inertia χ optimisation process for all thermal observa-tions, assuming a spherical shape model and 18 different pro- and retrogradespin-axes orientations (at latitudes ± ◦ , ± ◦ , ± ◦ , and arbitrary longitudes). χ -values and often show no clear minimum within the very largerange of thermal inertias. In addition to clear indications for the sense of rotation the χ -analysisalso points towards thermal inertias in the approximate range 400-1200 J m − s − . K − wherethe lowest χ -values are found. The 3 curves with lowest χ -minima are connected to ( λ SVecl , β SVecl ) = (90 ◦ , -60 ◦ ), (60 ◦ , -30 ◦ ), (0 ◦ , -90 ◦ ). On basis of our photometric data set and withoutusing additional visual light-curve information, it is apparently not possible to constrain thespin-axis orientation further within the retrograde domain.The connected effective radiometric diameter ( β SVecl = − ◦ ) is 0.31 ± ± χ -minima connected to the pole-solutions(90 ◦ , -60 ◦ ) and (60 ◦ , -30 ◦ ) are within this error range. Kaasalainen et al. (in press) published a shape model with a spin-vector solution basedon a large set of remote, disk-integrated photometric light-curve observations during the years2000 to 2004. The long time-line allowed to determine an accurate rotation period, a highquality pole solution, and a shape estimate. The shapes derived from light-curve inversiontechniques are reproducing the existing set of visual light-curves, but they do not have anabsolute size information connected to it. The Kaasalainen shape model for (25143) Itokawahas 1022 vertices and 2040 facets and it agreed well with the radar-based solution (Ostro et al.2005). The rotation parameters are β pole = − ◦ ± ◦ (retrograde sense of rotation), λ pole = 330 ◦ for the ecliptic latitude and longitude of the pole, and P = 12 . ± . γ = 0 . ◦ ) of this shape model is connected to a zero time ofT = 2451933 . χ -procedure (see M05 and Sect. 3.2) with the given Kaasalainen spin-vector for a range of thermal inertias. The primary goal was to determine the effective sizeof the scale-free Kaasalainen shape-model and to narrow down the possible thermal inertia ofItokawa’s surface. This time we also modified the TPM surface roughness to investigate theinfluence in the optimisation process (figure 4). The new minimum of 2.9 in the χ -calculations(lowest solid line in figure 4) is now significantly lower than in the case of a spherical shapemodel, indicating that the effects of the non-spherical shape are clearly dominating the χ -optimisation. But the χ -values are still relatively high and show that some data points are not9 ig. 3. The implementation of the Kaasalainen shape model with 2 040 facets (top) and the Gaskellshape model with 49 152 facets (bottom) for the epoch 01-Jul-2004 06:03 UT. For the temperature cal-culation a thermal inertia of 700 J m − s − . K − has been used. The viewing geometry is in the eclip-tic coordinate system as seen from Earth, projected on the sky. The Sun is at a phase angle of 54 ◦ .The asteroid’s apparent position (Earth-centred) was 312 ◦ ecliptic longitude and -48 ◦ ecliptic latitude. well matched. In a second round of χ -calculations we deselected all data which are marked astaken under bad weather conditions (labeled with (cid:63) in M05 and airmass > . χ minimum is at 0.9 and we obtained an excellent match between observed fluxes and modelpredictions. Our findings can be summarised as follows: (1) Additional surface roughness isneeded in the modelling to obtain acceptable χ -solutions. Without roughness (dotted linesin figure 4) the χ -minima are a factor 2-3 higher, mainly caused by the poor match to thedata at shortest wavelengths and smallest phase angles. (2) The surface roughness influencesthe calculations. It plays an important role at mid-IR wavelength where the thermal infraredemission peaks (see also M¨uller 2002): roughness variations cause a change in SED slopes andin second order also a change in absolute thermal fluxes, with different impact at different phaseangles. Depending on the surface roughness, the minima in figure 4 are shifting slightly and10 ig. 4. The thermal inertia χ -optimisation process for all thermal observations (solid lines) and a selectedsubset (dashed lines) assuming the shape model from Kaasalainen et al. (in press). The two dotted linesrepresent the corresponding results for a smooth surface without additional roughness. The 5 solid lines andthe 5 dashed lines are the results for different levels of roughness ranging from a relatively smooth surface( ρ =0.4, f=0.4) to an extremely rough surface completely covered by hemispherical craters ( ρ =1.0, f=1.0). therefore add to the thermal inertia uncertainty. (3) At a certain level of surface roughness it isnot possible anymore to distinguish between roughness influence and thermal inertia influence.More observational data closer to opposition would be needed to disentangle these competingsurface properties in the TPM. (4) The full dataset and also the selected high-quality datasetshow the χ -minima at similar thermal inertias: The most likely thermal inertia solutions arein the range 600-1100 J m − s − . K − (higher values are connected to higher levels of surfaceroughness and vice versa). (5) The quality limitations of our set of thermal observationsdetermine the χ -minima. A few low quality photometric observations dominate the finaluncertainties. (6) The corresponding effective radiometric diameter is 0.320 ± ± − s − . K − . These values are inexcellent agreement with the in-situ results. One result from the Hayabusa-mission is the highly accurate physical description of(25143) Itokawa. These shape models were produced by Robert Gaskell and available in dif-ferent resolutions from http://sbn.psi.edu/pds/resource/itokawashape.html . The fourresolutions correspond to 6( Q + 1) vertices and 12 Q facets, with Q = 64 , , , eff = 0.334 km of an equal volume sphere.11 able 4. Summary of the critical radiometrical properties from different sources. The numbers in bold face indicate thecurrent best values. Size/Shape colours & thermal params η or Remarks/Source D eff [m] geom. albedo p V Γ [Jm − s − . K − ] CommentsOstro et al. 2001 630( ± × ±
30) m — — radar 2001Sekiguchi et al. 2003 352 +28 − m 0.23 +0 . − . NEATM - η = 1.2 single N-bandOhba et al. 2003 a/b=2.1, b/c=1.7; triaxial ellipsoid — — light-curve inversionIsighuro et al. 2003 620( ± × ± × ±
30) m 0.35 ± - η = 1.1 M (cid:48) & N-bandKaasalainen et al. 2003 a/b=2.0, b/c=1.3; triaxial shape no variegation — light-curve inversionOstro et al. 2004 548 × ×
276 m; 358 ( ± × ×
288 m; 364 ( ± eff =280 m — Γ=350 multi-epoch M (cid:48) -/N-dataM¨uller et al. 2005 D eff =320 ±
30 m 0.19 +0 . − . Γ=750 ±
250 multi-epoch N-/Q-dataLowry et al. 2005 a/b > × × ± ± — — in-situThomas-Osip et al. 2008 a/b=1.9 ± ± V — V & NIR observationsGaskell et al. 2008 D eff = 334 m; highres. shape models — — Hayabusa/AMICABernardi et al. 2009 — H V & G-slope — V-band datathis work (Sect. 3.2) 310 ±
40 m; spin-axis estimate 0.30 ± ±
29 m 0.299 ± ± Γ = 500-900 in-situ shapeNotes. NEATM: Near-Earth Asteroid Thermal Model; FBM: free beaming parameter thermal model.
The true-size shape model allows now to investigate the influence of the aspect angle(defined such that it equals 0 ◦ when observing the North pole and it equals 180 ◦ when observingthe South pole of the object) on H-mag calculations and consequently also on the determinationof the geometric albedo (see also O’Rourke et al. 2012 on similar considerations for (21) Lutetia).Bernardi et al. (2009) measured V-magnitudes of Itokawa for a wide range of phaseangles and determined the corresponding mean light-curve values via a fit of synthetic light-curves (using the Gaskell shape model) to the observed incomplete light-curves. But dependingon the aspect angle, the light-curved averaged cross-section can vary between about 320 m and410 m! The relevant cross-sections (phase angles < ◦ ) for the H-mag determination in Bernardiet al. (2009) were all very small (324 ± +0 . − . mag is therefore only applicable for this smaller cross-section. Thecorresponding geometric albedo p V (relations in Bowell et al. 1989) is 0.31 ± V of 19.472 ± < ◦ ) were slightly smaller (just around 0.330 km) than the effective diameter of the Gaskellshape model (of an equal volume sphere). The corresponding geometric albedo p V is 0.28 ± ◦ and aspect angles from 50 to about 150 ◦ . Merging both datasets and considering theaspect angle limitations of the individual sets, we assign a geometric albedo of 0.29 ± of 1366 W m − ). The true, object-connectedH-mag, averaged over all aspect angles and connected to the average size of an equal volumesphere, is then H V =19.4 ± Similar to our analysis in Sect 3.2 and 3.3 we determined the size, albedo and thermalinertia information through our TPM implementation and using the “Gaskell” shape modelwith the absolute size as a free parameter. The corresponding χ -picture is very similar tofigure 4. We obtained minimum (reduced) χ -values around 3 for the full data-set and justbelow 1 for the high-quality sub-set of observations. The best match between observations andmodel predictions was found for an effective size of 0.332 ± ± − s − . K − at the χ -minimum and assuming an intermediate level of surface roughness. The radiometrical sizederived in this way is in excellent agreement with the true in-situ size. This validates ourmodel implementation and analysis technique. But similar to figure 4, we see a small shiftin thermal inertia when we modify the roughness level. An extremely rough surface (f=1.0, ρ =1.0: 100% of the surface covered by hemispherical craters with a surface slope r.m.s. of 1)requires slightly higher thermal inertias up to about 900 J m − s − . K − . A smoother surface(f=0.4, ρ =0.4: 40% of the surface covered by shallow craters with a surface slope r.m.s. of0.4) has to be combined with a lower thermal inertia (500 J m − s − . K − ) to obtain a goodmatch between model predictions and observed fluxes. This possible range for the thermalinertia can be translated into a mean grain radius of the surface regolith of about 21 +3 − mm(Gundlach & Blum 2013) which is in excellent agreement with the in-situ findings (Yano etal. 2006; Kitazato et al. 2008). The heat transport within the top-surface layers is thereforedominated by radiation effects and material properties play a very small role for the thermalbehaviour of Itokawa.We also tested thermal model solutions without adding any additional roughness. The χ -test produced an acceptable solution ( χ -minimum only about 15% higher than for thedefault roughness case), but the corresponding radiometric effective size was below 0.3 km,well outside the possible range. This confirms again that a certain roughness level is needed Active Cavity Radiometer Irradiance Monitor (ACRIM) total solar irradiance monitoring 1978 to present(Satellite observations of total solar irradiance); access date 2014-01-28;
13o explain the available observations, even in case of highly detailed and structured shapemodels. In fact, the surface features in the Gaskell model still belong to the “global shape”.They are still large in comparison to the thermal skin depth scales . Therefore it is neededto add an artificial roughness to account for the thermal beaming effect (Lagerros 1998b).This effect occurs mainly at centimetre scales, with small contributions coming from surfaceporosity on smaller scales (Hapke 1996; Lagerros 1998a). Multiple scattering of radiationincreases the total amount of solar radiation absorbed by the surface and rough surface elementsoriented towards the Sun become significantly hotter than a flat surface (Rozitis & Green 2011).Without such a “beaming model” on top of the Gaskell model it was not possible to find aconvincing radiometric solution for all observations simultaneously. Here we used (as before forthe spherical and Kaasalainen shape models) the beaming model concept developed by Lagerros(1997), with ρ , the r.m.s. of the surface slopes and f , the fraction of the surface covered bycraters. The beaming model produces a non-isotropic heat radiation, which is noticeable atphase angles close to opposition (see also figure 8). But it also influences the shape of the SEDin the mid-IR which is very relevant for our data set (e.g., M¨uller 2002). But more thermaldata closer to opposition would be needed to fully characterise surface roughness properties ofItokawa. We combined the Gaskell shape model (accepting the Gaskell size scale) now with ourderived thermal properties: an intermediate roughness level (f=0.7, ρ =0.6: 70% of the surfacecovered by craters with a surface slope r.m.s. of 0.6) and a thermal inertia of 700 J m − s − . K − .In figure 5 we present the ratios between observed fluxes and the corresponding TPM predictionsand show this ratios as a function of phase angle, wavelength, and rotational phase. These kindof plots are very sensitive to changes in the thermal properties (a full discussion of the influencesis given in M¨uller 2002): a wrong thermal inertia would lead to slopes in the phase-angle picture(top in figure 5) with large deviations from 1.0 at the largest phase angles, while wrong beamingparameters are dominating at the smallest phase angles close to opposition. The wavelengthpicture (bottom in figure 5) gives indications about emissivity variations and is strongly reactingto beaming parameter variations, especially at wavelengths shorter than the peak wavelength.The observation/TPM ratios also change with different aspect angles and the different setsof measurements are not easy to compare. Our optimum TPM solution seems to combinethe available observations without any obvious remaining trend in phase angle, wavelength orrotational phase.Nevertheless, there are individual outliers where observations and model predictionsdiffer significantly. Some of the data at phase angles above 100 ◦ (also visible in figure 5 middle The 49,152 facet shape model has an average facet dimension of ∼ ∼ ∼ ig. 5. All thermal observations divided by the corresponding TPM prediction as a function of phaseangle (top), as a function of rotational phase (middle) and as a function of wavelength (bottom), withthe zero rotational phase in the TPM setup defined at JD 2451934.40110 or 2001-Jan-24 21:37:35 UT). χ -analysis using the Gaskell shape model is almost identical to the analysis usingthe Kaasalainen shape model: We obtained very similar χ -values and also the derived thermalproperties agree very well. This shows that our analysis is limited by the number and quality ofthe thermal observation and not by shape information. The uncertainties are pure r.m.s.-valuesfrom the 25 individually derived radiometric solutions, but they reflect to a certain extent alsouncertainties in surface roughness and thermal inertia and the quality of the mid-IR photometricdata points.
4. Model Comparison and Predictions
Now, with the true shape model at hand, it is interesting to compare thermal light-curves from the Kaasalainen shape model with the light-curves from the Gaskell shape model.In figure 6 we calculated for one full rotation period with a starting time 01-Jul-2004 at 00:00UT the visual light-curve (in relative magnitudes), based on the Gaskell shape-model (top),and the thermal light-curves (bottom) at different wavelengths (8.73, 10.68, 12.35, 17.72 µ m).At visible wavelength the Kaasalainen-shape model matches very well the existing light-curves(Kaasalainen et al. in press; ˇDurech et al. 2008). In the mid-IR thermal range the results of theKaasalainen shape model follows for large fractions of the light-curve from the Gaskell model.But at specific rotational phases the differences are significant. The Kaasalainen shape modelproduces a larger peak-to-peak thermal light-curve amplitude, nicely visible in the Q-band at17.72 µ m curve and it also produces bumps and sharp edges at certain phases.The relatively large facets in the Kaasalainen shape model cause artificial structures inthe predicted thermal light-curve while the Gaskell model produces astonishingly smooth andregular curves. In the given observing set there are also severe differences in the peak-to-peak16 ig. 6. Top: The visual light-curve in relative magnitudes, based on the Gaskellshape-model.
Bottom:
A comparison of thermal light-curves produced with theKaasalainen shape model (dashed lines) and the Gaskell shape model (solid lines)for the epoch 01-Jul-2004 when the asteroid was see at phase angles of 54-55 ◦ . variations from both models (about 10-20%, depending on the wavelength), but the deviationsvary not only with wavelength, but also with aspect angle. The minima in the thermal light-curve are also broader than for the visual light-curve. This is an effect of the object’s shape ata given rotation angle combined with the high thermal inertia which smoothes out the rapidchanges of illuminated surface areas. It can be concluded that the Kaasalainen shape model,although it was derived from light-curve observations and matches nicely visual light-curvesfor a wide range of observing geometries, still has shortcomings in the context of thermallight-curves. Or in other words: the light-curve inversion technique might benefit from usingthermal light-curves and the resulting shape models and spin-axis orientations would comecloser to reality. The Gaskell shape model in combination with the derived and validated thermophysicalproperties allows now to do more generalised studies. What can be learnt from thermal spectraor light-curve measurements at different wavelengths? How does the opposition effect look likeat thermal wavelengths? What are the key observing geometries for successful radiometriccalculations?Figure 7 (top) shows model predictions (now only using the Gaskell shape model) for awavelength range from 5 to 50 µ m. Both SEDs at intermediate roughness level were normalisedto 1.0 at the thermal emission peak wavelengths. Nevertheless, one can see that the SED atsmall phase angles (dashed lines) are higher (in the longer wavelength range) than the SEDs atlarger phase angle (solid lines). The surface roughness and its thermal-infrared beaming effectenhance the observed thermal emission at low phase angles. This is then balanced out by a17 ig. 7. The Gaskell shape model combined with thermal properties: Thermal effects as a function ofwavelength.
Top:
The normalised SEDs for observations close to opposition (dashed lines) and at 54 ◦ phase angle (solid lines) for a thermal inertia of Γ=1000 J m − s − . K − . The strong roughness influenceat small phase angles is clearly visible (thermal opposition effect). Bottom:
The thermal light-curve(LC) amplitudes (peak-to-peak in [%] of the absolute thermal flux) for the observing constellation on01/Jul/2004 (54 ◦ phase angle). For the “low thermal inertia” case we used 15 J m − s − . K − (typical valuefor large main-belt asteroids; M¨uller et al. 1999) and for the “high thermal inertia” 1000 J m − s − . K − .The pure shape-caused light-curve amplitude (as seen in visual light) is indicated by the horizontal line.The peak-to-peak amplitude decreases significantly for longer wavelengths and for higher thermal inertias. ◦ ) as a function of wavelength fortwo values of the thermal inertia. The “low thermal inertia” case represents typical main-belt values (M¨uller et al. 1999) caused by a very well insulating dust regolith on the surface(15 J m − s − . K − ). For comparison, the Moon has a thermal inertia of 39 J m − s − . K − (Keihm 1984). The “high thermal inertia” case corresponds to our best solution for the ther-mal properties of (25143) Itokawa. In the given geometry the peak-to-peak brightness variationduring one full rotational period is changing dramatically with wavelength. At mid-IR theamplitude in the “low thermal inertia” case can even exceed the pure shape-caused brightnessvariation! The thermal light-curve amplitude decreases by more than a factor of 2 from mid-IRto far-IR wavelengths and this effect is almost independent of phase angle. In the high thermalinertia case (Itokawa) the overall values are significantly smaller (much smaller than the shape-introduced amplitude) and also the change with wavelength is smaller. Close to opposition thelight-curve amplitude behaviour in the high thermal inertia case is more complex and does notshow a clear trend with wavelength anymore. Overall, the peak-to-peak thermal light-curveamplitude decreases for higher thermal inertias and at longer wavelengths. Uncertainties in thelight-curve amplitudes due to stronger influences of surface roughness increase for observationsclose to opposition.In figure 8 we averaged the flux predictions over a complete rotation and looked at thechanges with phase angle covering large periods before and after opposition. The predictionshave been normalised at α = 0 ◦ and for default roughness. The flux change is dominated bythe distance change between Earth and asteroid. The error bars indicate the uncertaintiesdue to thermal inertia, the dashed lines give the boundaries for very low and high beamingvalues, i.e. low and high surface roughness. At small phase angles there is another effectvisible: the thermal opposition or beaming effect. At small phase angles it is possible tosee warmer temperatures inside the small-scale surface structures (modelled by the crater-like19 ig. 8. TPM/Gaskell predictions for Itokawa-opposition in 2011. The corresponding times are:03-Apr-2011: -15.0 ◦ (trailing the Sun), 13-Jun-2011: 0.1 ◦ , 27-Aug-2011: +15.0 ◦ (leading the Sun). Top:
Normalised (by the asteroid distance to Sun and Earth) 10 µ m fluxes as a function of phaseangle. Uncertainties due to thermal inertia (range from 500-1000 J m − s − . K − ) are given with er-ror bars, uncertainties due to roughness (range for ( ρ ,f) from (0.4,0.4) to (0.9,0.9)) are indicated bythe dashed lines. The opposition “peak” at small phase angles can be seen, as well as the influenceof surface roughness on the peak-hight. Bottom: the peak-to-peak light-curve variation at 10 µ m inpercent as a function of phase angle. The relative light-curve amplitude is significantly smaller closeto opposition. The influences of roughness and thermal inertia change slightly with phase angle. > ◦ ) and close to the emission peak (in the mid-IR for NEAs) are providing themost stringent constraints on the thermal inertia: the light-curve amplitude is maximal andthe beaming influence is very small.We also looked into synthetic thermal light-curves for the Akari observing epochs andusing the Gaskell shape model. Figure 9 shows these light-curves at 4.1 (top) and 15.0 µ m(bottom) for a wide range of thermal inertias. The TPM code produces absolute fluxes whichcompare very well with the derived Akari flux densities which are over-plotted in the figures.Our calculations show that the thermal light-curves change dramatically with changing thermalinertia. At 4.1 µ m the light curve shape goes from a relatively smooth curve with two differentpeak levels (low thermal inertia) to a much more structured curve with sharp turns (highthermal inertia). The amplitude is not very much affected, but the delay times of the curvesincrease with increasing thermal inertia. In the high thermal inertia case the thermal emissionpeaks happen much later and delay times of up to 2 hours are found. At these short wavelengthsit is mainly the hottest sub-solar terrains which dominate the observed fluxes and flux changes.But due to the roughness-thermal inertia degeneracy (e.g. Rozitis & Green 2011) the short-wavelength observations are not ideal for deriving highly reliable size-albedo solutions. However,the shift of the thermal curves with respect to a purely shape-driven light-curve are closelyconnected to the object’s thermal inertia. Monitoring light-curve changes over part of therotation period is therefore the recommended approach at the short wavelengths. At 15.0 µ mthe situation is different: the light-curves are in general much smoother and their shapes donot change much with thermal inertia, and the peak thermal emission delay times are smallerthan at shorter wavelengths. These measurements usually lead to more robust size-albedodue to the reduced influence of the roughness-thermal inertia degeneracy (e.g. M¨uller 2002).Accessing the object’s thermal inertia is more difficult: either one has to measure the thermallight-curve amplitude (which is substantially decreasing with increasing thermal inertia) or onehas to combine the measurements with observations at different phase angles. It is also worthto note here that the thermal light-curve shapes, amplitudes, delay times for a given objectalso depend on the phase angle. The Akari measurements were taken at a phase angle of 73 ◦ where the illuminated (hot) part of the surface dominates the short wavelength observations,while at longer wavelengths there are also flux contributions from the warm, non-illuminatedparts which have just rotated out of the Sun. 21 ig. 9. TPM/Gaskell thermal light-curve predictions for Itokawa during the Akari ob-serving epochs for a wide range of thermal inertias.
Top: calculated thermallight-curves at 4.1 µ m together with the observed Akari data point. Bottom: calcu-lated thermal light-curves at 15.0 µ m together with the observed Akari data point. . Discussion The interpretation of the 30 thermal infrared observations by using only a sphericalshape model (Sect. 3.2) was very successful: the radiometric effective size prediction lies withina few percent of the true, in-situ value, the radiometric albedo prediction agrees within theerror-bars with Bernardi et al. (2009), the retrograde sense of rotation was clearly favoured inthe optimisation process, and thermal inertia values in the range 500-1500 J m − s − . K − arethe most likely ones.But the calculations benefited from the proximity of the assumed spin axis orientationwith the true orientation and from using a realistic rotation period. If an object’s rotationperiod is very different from the TPM assumptions and/or the spin vector is far away from theecliptic pole direction the TPM predictions will be less reliable. Also the number of thermalobservations, their distribution in phase angle space, in wavelength space, in rotational phasespace, in aspect angle space and observing geometry influence the quality of the TPM outcome.Overall, the reduced χ -values are relatively high, indicating that the spherical shape isnot always allowing to produce model predictions within the given observational errors. Themodel predictions for the few cases of extreme cross-sections dominate the overall χ -sums.But here the observations are statistically distributed over rotational phases and aspect anglesand the χ -values remain sensitive to basic object properties like thermal inertia and sense ofrotation. If only very few thermal observations would be available, even the distinction betweenpro- and retrograde solutions could fail for cases where shape effects and/or phase angle effectsinterfere with thermal signatures in the terminator region. M¨uller (2002) demonstrated thecapabilities and limitations of the “radiometric method” to determine the sense of rotation viavery spherical shape models. Based on a sample of 9 main-belt asteroids with comparable setsof thermal observations before and after opposition, the thermal data indicated the correctsense of rotation for 8 objects. In one case it failed, very likely due to shape and cross-sectioneffects and data sets which did not cover the full rotation periods.Nevertheless, our extremely simplified model approach shows the large potential of theradiometric technique: Assuming a spherical shape in combination with a realistic rotationperiod and a range of spin-axes it is possible to derive the sense of rotation and very accu-rate values for the effective size. But the key to robust results is the analysis of all thermalobservations simultaneously, combined with all available information from visual photometryand light-curves observations. The very accurate radiometric diameter prediction has to beseen in comparison with the radar techniques: (Ostro et al. 2004; Ostro et al. 2005) useddelay-Doppler images of (25143) Itokawa obtained at Arecibo and Goldstone to predict a sizeof 594 × ×
288 m ( ± ◦ . The new H-magnitude differs by 0.5 mag from theones used in M05. This influences the albedo considerably (see discussion in M05) while theradiometric diameter is less affected. The difference in albedo (0.19 +0 . − . in M05 and 0.299 ± . First, by usinga spherical shape model, then by a modified shape model which would match the existing visuallight-curves in amplitude and in phase. The quality of the resulting radiometric properties(diameter, albedo, thermal inertia, ...) suffered from an unknown spin vector orientation atthe time of the calculations. Nevertheless, the example of (162173) 1999 JU shows nicely thepotential and the limitation of applying this method without knowing the precise shape northe spin vector orientation. Using the shape model (together with the spin-vector orientation) from light-curve in-version techniques led to a radiometric diameter which was within 2% of the true in-situ result(M05; Fujiwara et al. 2006). The thermal observations put very strong constraints on theradiometric diameter. If, in addition, the H-G values are reliable, i.e., representing light-curve-averaged properties, or if simultaneous V-magnitudes are available, then both, the effectivediameter and the albedo can be derived with high accuracy. Alternatively, the size informationfrom radar techniques or occultation measurements can be tested with TPM calculations: Isit possible to find realistic thermal properties to explain thermal measurements and the inde-pendent size values simultaneously? M05 tried such an approach with Itokawa’s radar size, butthey could not find any acceptable match with the observed fluxes and they concluded that theradar size must be too large.Our optimisation process clearly has limitations: Figure 4 demonstrates that the bestsolutions from using all 30 data points were strongly affected by a few less reliable observations.Thermal observations suffer in some cases from poor absolute calibration (or badly documentedcalibration), from high humidity weather conditions and/or high air-masses, or from uncertain-ties in the definition of band-passes with errors in the colour correction terms. The plots withobservations/model ratios as a function of wavelength, phase angle and rotational phase re-vealed in many cases the outliers, at least if a high quality shape model (covering many aspectangles) is available.Another limiting factor is the uncertainty in surface roughness. An artificial roughness24odel is needed to explain typical mid-IR asteroid spectra (e.g., Barucci et al. 2002; M¨uller &Blommaert 2004; Dotto et al. 2000; M¨uller & Lagerros 2002), but the precise values or the inter-pretation of roughness is more complex: The thermal infrared observations are disk-integratedobservations, averaging over very diverse surface regions. The resulting disk-integrated ther-mal properties, especially the roughness, are therefore of limited significance for characterisingindividual regions on the surface. Our “default” roughness model explains the existing ob-servations very well, although it works in certain observing geometries better than in others,but more observations close to opposition (see figure 7) or at shorter wavelengths (e.g. M¨uller2002 or Rozitis & Green 2011) would allow to narrow down the range in the possible rough-ness parameter space or to even to distinguish between hemispheres or large surface regionswith different roughness properties. It should also be noted here that the roughness influencesradiometric size and albedo solutions strongly if thermal infrared observations are only takenclose to opposition and/or at short wavelengths in the Wien-part of the thermal emission.Overall, the step from spherical to Kaasalainen shape model changed only slightly theeffective diameter, geometric albedo and thermal inertia values, but the reliability improvedsignificantly: The χ -minima shrank from ∼ The availability of the full, high resolution shape model with spin-vector orientation andabsolute size information (Demura et al. 2006) allows to obtain confidence in the radiometrictechnique: Using only the shape model and spin-vector orientation in combination with theset of thermal infrared observations we found the best fit to all data at a thermal inertiaof 700 J m − s − . K − and under the assumption of a “default roughness”. The correspondingradiometric effective diameter agrees then nicely with the true value. The radiometric diameteritself has a formal uncertainty of ±
8% when taking the r.m.s.-residuum from the 25 observationsor 8 / √
25 = ± ±
15% or a statistical error of the mean of about ± ± ± ± − s − . K − and regolith-covered bodies, likethe Moon or large main-belt asteroids. (25143) Itokawa has an intermediate thermal inertia,possibly indicating a rubble pile structure where seismic waves reorganise the body’s interiorand the surface frequently and the formation of a thick regolith (with thermal inertias below100 J m − s − . K − ) is hampered. The reason for having very little fine regolith could also berelated to the weak gravity on small, low density objects. Small asteroids tend to have ingeneral higher thermal inertias (Delb´o et al. 2007), but there are exceptions (e.g., M¨uller et al.2004).
6. Conclusions (25143) Itokawa is an extremely important test case for the validation process of ther-mophysical model techniques. The existing mid-IR observations allowed to evaluate the possi-bilities and limitations of different levels of complexity within our TPM implementation. Evenin cases where very little is known about shape and spin behaviour it allows to derive reliableproperties, like the effective size, the albedo and thermal properties without using any artificialmodel fudge factors. But the outcome is tightly connected to the availability and quality ofthermal infrared observations. Ideally, the observations should (i) cover a sufficient wavelengthrange around the emission peak (several photometric bands or mid-IR spectra); (ii) includemeasurements before and after opposition; (iii) cover a large phase angle range, including mea-surements close to opposition; and, (iv) include a significant range of rotational phases and/orsubstantial parts of the thermal light-curve. It is also important to note here that some of themid-IR bands might be affected by silicate emission (e.g. Emery et al. 2006), but the effect on26ize-albedo solutions can only be estimated if full thermal spectra are available.Our analysis showed that for (25143) Itokawa the interpretation of surface roughnessproperties is limited mainly due to the lack observations close to opposition. Nevertheless,Itokawa’s size, albedo and thermal inertia have been derived with unprecedented accuracy byonly using remote, disk-integrated observations: a shape model from standard light-curve inver-sion technique and thermal infrared observations from ground and from AKARI. The optimumradiometric size agreed within 2% with the true value derived from Hayabusa measurements.The TPM predictions using the true in-situ shape model showed: (i) that the shape mod-els from light-curve inversion techniques produce artefacts in thermal light-curves and that lowthermal inertia object (e.g., large main-belt asteroids) can have light-curve amplitudes exceed-ing the pure shape-introduced values; (ii) that the SEDs taken close to opposition are stronglyinfluenced by properties of the surface roughness (figure 7, top) leading to a strong degeneracybetween roughness and thermal inertia effects; (iii) that there exists a thermal opposition ef-fect and how it looks like (figure 8, top), (iv) how the thermal light-curve amplitude changeswith phase angle (figure 8, bottom); (v) that the thermal light-curve amplitudes decrease withwavelength and for higher thermal inertias (figure 7, bottom); (vi) that thermal light-curvedelay times increase with thermal inertia and decrease for longer wavelengths (figure 9); (vii)that the thermal inertia and the sense of rotation play a big role when interpreting thermalinfrared observations at large phase angles. (viii) that even the high resolution Gaskell shapemodel still require an additional small-scale roughness to explain the observed infrared fluxes.Our findings are supported by TPM analysis of other spacecraft target asteroids. In thiscontext one should mention some of the recent studies on (4) Vesta (M¨uller & Lagerros 1998;M¨uller & Lagerros 2002; Leyrat et al. 2012; Keihm et al. 2013), (21) Lutetia (Mueller et al.2006; Lamy et al. 2010; O’Rourke et al. 2012; Keihm et al. 2012), (433) Eros (Mueller 2007), and(2867) Steins (Lamy et al. 2008; Groussin et al. 2011; Leyrat et al. 2011). In all these cases theradiometrically derived properties are remarkably consistent with the spacecraft investigationsdocumented by Russell et al. (2012) for (4) Vesta, by Sierks et al. (2011) for (21) Lutetia, byThomas et al. (2002) for (433) Eros, and by Keller et al. (2010) for (2867) Steins. The TPMradiometric technique is very powerful in deriving highly reliable absolute sizes, albedos, andthermal inertias from remote disk-integrated thermal measurements. But each of the availabledata points has to be considered in its true illumination and observing geometry. In thisway it is also possible to extract information on the object’s spin properties as well as on itsshape, especially for objects where standard lightcurve inversion techniques have difficulties todetermine these parameters.The validated model techniques can easily be used for other targets, including near-Earthand main-belt asteroids, trans-Neptunian objects or inactive cometary nuclei. The plots andfigures can also be used to optimize observing strategies to exploit the full TPM capabilities.The key ingredient for the full exploitation of thermophysical model techniques and for the27etermination of reliable object properties is the availability of well-selected and well-calibratedthermal infrared observations covering many aspect angles.S. Hasegawa was supported by Space Plasma Laboratory, ISAS, JAXA. We would like tothank Johan Lagerros for very useful discussions and Robert Gaskell for providing the necessarydocumentation for the implementation of Itokawa’s shape model.
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