deliverable title according to ga · 1.0 8 march 2016 interim release to project 1.1 23 september...

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 640351.

NEOShield-2 Science and Technology for Near-Earth Object Impact Prevention

Grant agreement no: 640351 Project Start: 1 March 2015

Project Coordinator Airbus Defence and Space DE Project Duration: 31 Months

WP 11, Task 11.1 Deliverable D11.2

Report on data analyses and mitigation-relevant NEO physical properties

WP Leader DLR Task Leader DLR Due date M14, 30 Apr 2016 (final M29, 31 July 2017) Delivery date Original: 13.07.2017, revised: 10.08.2017 Issue 3.0 Editor (authors) Alan Harris, Line Drube Contributors Alan Harris, Line Drube Verified by Alan Harris, Line Drube Document Type R Dissemination Level PU

The NEOShield-2 Consortium consists of: Airbus Defence and Space GmbH (Project Coordinator) ADS-DE Germany Deutsches Zentrum für Luft- und Raumfahrt e.V. DLR Germany Airbus Defence and Space SAS ADS-FR France Airbus Defence and Space Ltd ADS-UK United Kingdom Centre National de la Recherche Scientifique CNRS France DEIMOS Space Sociedad Limitada Unipersonal DMS Spain Fraunhofer-Gesellschaft zur Förderung der angewandten Forschung e.V. EMI Germany GMV Aerospace and Defence SA Unipersonal GMV Spain Istituto Nazionale di Astrofisica INAF Italy Observatoire de Paris OBSPM France The Queen’s University of Belfast QUB United Kingdom

NEOShield-2 D11.2 Title: NEO Physical Properties Issue, Date 3.0, 10.08.2017 /42

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Change Record

Issue Date Section, Page Description of Change

0.1 23 October 2015 First draft

0.2 24 February 2016 Advanced draft interim version

1.0 8 March 2016 Interim release to project

1.1 23 September 2016 3.4.3, p.31 Additional material inserted

1.2 5 October 2016 Additional material inserted

1.3 7 June 2017 3.5, p.33 Additional material inserted

1.4 15 June 2017 All Minor corrections

1.5 27 June 2017 3.5.2, p.38 Final section completed

1.6 13 July 2017 Draft final version for internal project review

2.0 31 July 2017 Final submitted version

2.5 8 August 2017 40, 41 Responses to project internal review

3.0 10 August 2017 All Minor corrections and clarifications. Final revised version.


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Table of Contents

1 Introduction ............................................................................................................................................. 4

1.1 Scope .................................................................................................................................................. 4

1.2 List of Abbreviations .................................................................................................................... 4

1.3 Applicable Documents ................................................................................................................ 4

1.4 Reference Documents ................................................................................................................. 4

2 The Importance of Physical Characterisation for Planetary Defense ................................. 8

2.1 In-situ characterisation .............................................................................................................. 8

2.2 Astronomical observations ....................................................................................................... 8

2.3 Review of results from NEOShield-1 and relevant thermal physics ........................ 10

3 Further analysis of published infrared observations of asteroids .................................... 12

3.1 The dependence of η on solar aspect angle ...................................................................... 12

3.2 The NEATM beaming parameter and thermal inertia .................................................. 14

3.3 A NEATM-based thermal-inertia estimator for near-Earth asteroids .................... 19

3.4 Application of the NEATM-based asteroid thermal-inertia estimator ................... 26

3.4.1 Main-belt asteroids .............................................................................................................................. 26

3.4.2 Near-Earth objects ............................................................................................................................... 30

3.4.3 Thermal inertia and rotation rate .................................................................................................. 32

3.5 Further investigations with the WISE data ....................................................................... 34

3.5.1 Using default values for the solar aspect angle ........................................................................ 34

3.5.2 Thermal inertia estimates of NEOShield-2 targets ................................................................. 38

4 Conclusions ............................................................................................................................................ 39

5 Appendix ................................................................................................................................................. 41


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1 Introduction

1.1 Scope The principal aim of this task is to analyze available observational data, from this project, and available complementary data from others (e.g. NEOWISE), to provide mitigation-relevant statistical information on the population of small NEOs as a whole, and therefore the impact hazard. The results should provide a means to constrain the mitigation-relevant properties of a particular NEO on the basis of statistical knowledge of NEO physical properties in general. Data from different observational techniques (e.g. optical, infrared, radar, and data from space missions) are gathered and combined. Correlations between diverse observed asteroid parameters are explored to provide further insight into relationships between thermal properties, mineralogy, albedo, surface characteristics, spin rate, etc. Such relationships will also aid in identifying and correcting erroneous information in current databases, such as taxonomic types, spin vectors, albedos, absolute magnitudes, etc. This work is a continuation of that carried out in NEOShield-1 (WP2, as described in D2.1) taking account of newly available data.

1.2 List of Abbreviations AD Applicable Document

AU Astronomical Unit

IRAS Infrared Astronomical Satellite

MBA Main-belt Asteroid

NEO Near-Earth Object

RD Reference Document

TP Thermal Parameter

WISE Wide-field Infrared Survey Explorer

WP Work Package

1.3 Applicable Documents [AD1] NEOShield-2: Science and Technology for Near-Earth Object Impact Prevention, Grant

Agreement no. 640351, 28.10.2014.

[AD2] Harris, A. W., Drube, L., 2014, Report on frequency of mitigation-relevant properties in the NEO population, Harris, A. W., Drube, L., 2014, NEOShield Deliverable 2.1.

1.4 Reference Documents [RD1] Bowell E., Hapke B., Domingue D., Lumme K., Peltoniemi J., and Harris A. W., 1989,

Application of photometric models to asteroids. In Asteroids II (R. P. Binzel, et al., eds.), pp. 524–556. Univ. of Arizona Press, Tucson.

[RD2] Brown, R. H., 1985, Ellipsoidal geometry in asteroid thermal models: The standard radiometric model. Icarus, 64, 53–63.

[RD3] Bus, S. J., Binzel, R. P., 2002, Phase II of the Small Main-belt Asteroid Spectroscopy Survey: A feature-based taxonomy. Icarus, 158, 146–177.


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[RD4] Busch, M. W. et al., 2008, Physical properties of near-Earth Asteroid (33342) 1998 WT24. Icarus, 195, 614–621.

[RD5] Campins, H., Emery, J. P., Kelley, M., Fernández, Y., Licandro, J., Delbó, M., Barucci, A., Dotto, E., 2009, Spitzer observations of spacecraft target 162173 (1999 JU3). Astron. Astrophys, 503, L17–L20.

[RD6] Čapek, D. and Vokrouhlický, D., 2004, The YORP effect with finite thermal conductivity. Icarus, 172, 526–536.

[RD7] Chesley, S. R., et al., 2003, Direct detection of the Yarkovsky effect by radar ranging to asteroid 6489 Golevka. Science, 302, 1739–1742.

[RD8] Delbo, M., Mueller, M., Emery, J. P., Rozitis, B, Capria, M.-T., 2015, Asteroid thermophysical modeling. In Asteroids IV (P. Michel et al., eds.), pp. 107–128. Univ. of Arizona Press, Tucson. DOI: 10.2458/azu_uapress_9780816532131-ch006.

[RD9] Delbo, M., Walsh, K., Mueller, M., Harris, A. W., Howell, E. S., 2011, The cool surfaces of binary near-Earth asteroids. Icarus, 212, 138–148.

[RD10] Emery, J. P., et al., 2014, Thermal infrared observations and thermophysical characterization of OSIRIS-REx target asteroid (101955) Bennu. Icarus, 234, 17–35.

[RD11] Golombek, M. P., Jakosky, B. M., and Mellon, M. T., 2003, Thermal inertia of rocks and rock populations and implications for landing hazards on Mars. In First Landing Site Workshop for the 2003 Mars Exploration Rovers, NASA Ames Research Center, Mountain View, CA., Abstract 9017.

[RD12] Gulkis, S., et al., 2012, Continuum and spectroscopic observations of asteroid (21) Lutetia at millimeter and submillimeter wavelengths with the MIRO instrument on the Rosetta spacecraft. Plan. and Space Sci., 66, 31–42, DOI: 10.1016/j.pss.2011.12.004.

[RD13] Hanus, J. et al., 2011, A study of asteroid pole-latitude distribution based on an extended set of shape models derived by the lightcurve inversion method. Astron. Astrophys., 530, A134.

[RD14] Harris A. W., Boslough M., Chapman C. R., Drube L., Michel P., Harris A. W., 2015, Asteroid impacts and modern civilization: Can we prevent a catastrophe? In Asteroids IV (P. Michel et al., eds.), pp. 833–852. Univ. of Arizona Press, Tucson. DOI: 10.2458/azu_uapress_9780816532131-ch042.

[RD15] Harris, A. W. and Davies, J. K., 1999, Physical Characteristics of near-Earth asteroids from thermal infrared spectrophotometry. Icarus, 142, 464–475.

[RD16] Harris A. W. and Drube L., 2014, How to find metal-rich asteroids. Astrophys. J. Lett., 785, L4, DOI: 10.1088/2041-8205/785/1/L4.

[RD17] Harris A. W. and Drube L., 2016, Thermal tomography of asteroid surface structure. Astrophys. J., 832:127, DOI: 10.3847/0004-637X/832/2/127.

[RD18] Harris, A. W., 1998, A thermal model for near-Earth asteroids. Icarus, 131, 291–301.

[RD19] Harris, A. W., Mueller, M., Delbo’, M., Bus, S. J., 2005, The surface properties of small asteroids: Peculiar Betulia—A case study. Icarus, 179, 95–108.

[RD20] Harris, A. W., Mueller, M., Delbo’, M., Bus, S. J., 2007, Physical characterization of the potentially hazardous high-albedo asteroid (33342) 1998 WT24 from thermal-infrared observations. Icarus, 188, 414–424.

[RD21] Hasegawa, S., et al., 2008, Albedo, size, and surface characteristics of Hayabusa-2 sample-return target 162173 1999 JU3 from AKARI and Subaru Observations. Publ. Astron. Soc. Japan, 60, S399–S405.


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[RD22] Keihm, S. J. and Langseth, M. G., 1975, Lunar microwave brightness temperature observations reevaluated in the light of Apollo program findings. Icarus, 24, 211–230.

[RD23] Lagerkvist, C.-I., et al., 1998, Physical studies of asteroids XXXIII. The spin rate of M-type asteroids. Astron. Astrophys., 131, 55–62.

[RD24] Lellouch, E., et al., 2013, “TNOs are Cool”: A survey of the trans-Neptunian region. Astron. Astrophys., 557, A60.

[RD25] Magri, C., Nolan, M. C., Ostro, S. J., Giorgini, J. D., 2007, A radar survey of main-belt asteroids: Arecibo observations of 55 objects during 1999-2003. Icarus, 186, 126–151.

[RD26] Mainzer, A. et al., 2011a, Preliminary results from NEOWISE: an enhancement to the Wide-field Infrared Survey Explorer for solar system science. Astrophys. J., 731:53 (13pp).

[RD27] Mainzer, A. et al., 2011b, NEOWISE observations of near-Earth objects: preliminary results. Astrophys. J., 743:156 (17pp).

[RD28] Mainzer, A. et al., 2014, The population of tiny near-Earth objects observed by NEOWISE. Astrophys. J., 784:110 (7pp).

[RD29] Mainzer, A. et al., 2016, NEOWISE Diameters and Albedos V1.0. EAR-A-COMPIL-5-NEOWISEDIAM-V1.0. NASA Planetary Data System, 2016. https://sbn.psi.edu/pds/resource/neowisediam.html

[RD30] Masiero, J. R. et al., 2011, Main belt asteroids with WISE/NEOWISE. I. Preliminary albedos and diameters. Astrophys. J., 741:68 (20pp).

[RD31] Masiero, J. R., Gray, T., Mainzer, A. K., Nugent, C. R., Bauer, J. M., Stevenson, R., Sonnett, S., 2014, Main-belt asteroids with WISE/NEOWISE: near-infrared albedos. Astrophys. J., 791:121 (11pp).

[RD32] Müller, T. G., et al., 2014, Thermal infrared observations of asteroid (99942) Apophis with Herschel. Astron. and Astrophys., 566, A22.

[RD33] Müller, T. G., et al., 2013, Physical properties of asteroid 308635 (2005 YU55) derived from multi-instrument infrared observations during a very close Earth approach. Astron Astrophys, 558, A97.

[RD34] Müller, T. G., Sekiguchi, T., Kaasalainen, M., Abe, M., Hasegawa, S., 2005, Thermal infrared observations of the Hayabusa spacecraft target asteroid 25143 Itokawa. Astron. and Astrophys., 443, 347–355.

[RD35] Pravec, P., Harris, A. W., Michalowski, T., 2002, Asteroid rotations. In Asteroids III (W. F. Bottke et al., eds.), pp. 113–122. Univ. of Arizona Press, Tucson. http://www.lpi.usra.edu/books/AsteroidsIII/pdf/3016.pdf

[RD36] Rivkin, A. S., Howell, E. S., Lebofsky, L. A., Clark, B. E., Britt, D. T., 2000, The nature of M-class asteroids from 3-μm observations. Icarus, 145, 351–368.

[RD37] Shepard, M. K., et al., 2010, A radar survey of M- and X-class asteroids. II Summary and synthesis. Icarus, 208, 221–237.

[RD38] Shepard, M. K., et al., 2012, Radar observations of seven X/M-class main-belt asteroids. 43rd Lunar and Planetary Science Conference, 1228.

[RD39] Shepard, M. K., et al., 2015, A radar survey of M- and X class asteroids. III. Insights into their composition, hydration state, & structure. Icarus, 245, 38–55.

[RD40] Spencer, J. R., Lebofsky, L. A., Sykes, M., 1989, Systematic biases in radiometric diameter determinations. Icarus, 78, 337–354.

[RD41] Tedesco, E. F., Noah, P. V., Noah, M., Price, S. D., 2002, The supplemental IRAS Minor Planet Survey. Astron. J., 123, 1056–1085.

https://sbn.psi.edu/pds/resource/neowisediam.html

http://www.lpi.usra.edu/books/AsteroidsIII/pdf/3016.pdf


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[RD42] Tholen D. J., 1984, Asteroid taxonomy from cluster analysis of photometry. Ph.D. thesis, Univ. of Arizona.

[RD43] Warner, B. D., Harris, A. W., Pravec, P., 2009, The asteroid lightcurve database. Icarus, 202, 134–146. Updated 2015 December 7. http://www.MinorPlanet.info/lightcurvedatabase.html

[RD44] Wesselink, A. J., 1948, Heat conductivity and nature of the lunar surface material. Bull. Astron. Inst. Neth., 10, 351–363.

[RD45] Wolters, S. D., Rozitis, B., Duddy, S. R., Lowry, S. C., Green, S. F., Snodgrass, C., Hainaut, O. R., Weissman, P., 2011, Physical characterization of low delta-V asteroid (175706) 1996 FG3. Mon. Not. R. Astron. Soc., 418, 1246–1257.

http://www.minorplanet.info/lightcurvedatabase.html


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2 The Importance of Physical Characterisation for Planetary Defense

2.1 In-situ characterisation Mitigation-relevant physical characterization is an important pre-requisite to reliable estimates of the effects of an impact on the ground, and the design of effective deflection missions. In-situ characterization via rendezvous and fly-by missions contributes important ground truth to complement the growing archives of astronomical data on NEOs (see below). The Near-Earth Asteroid Rendezvous - Shoemaker spacecraft (NEAR Shoemaker), named in honor of planetary scientist Gene Shoemaker, was the first rendezvous mission to a near-Earth asteroid. NEAR Shoemaker was launched from Cape Canaveral, Florida, in February 1996, on a 3 year journey to asteroid (433) Eros. NEAR orbited Eros for one year in 2000–2001. It provided the first detailed characterization of a NEO’s chemical and physical properties. The objective was to study Eros’ relationship to meteorites, the nature of its surface and collisional history as well as aspects of its interior state and structure. The Japanese Hayabusa spacecraft was launched in 2003 and started its 3-month rendezvous with the NEO Itokawa in 2005. At the end of this period the spacecraft collected a sample from the surface of Itokawa and returned it back to Earth, where the sample capsule containing ~1500 grains (mostly in the 10-50 micrometers size range) landed in Australia in 2010.

While their primary aim is to contribute to our understanding of solar system history, sample-return missions to asteroids also provide valuable information for planetary defense. Examples of such missions currently underway are Hayabusa 2 (Japan, with French and German participation) and OSIRIS-Rex (US with Canadian and French participation). The Hayabusa mission was the first mission to return samples of an asteroid to Earth. The micrometer-sized grains returned by Hayabusa from the NEA (25143) Itokawa allowed a direct link to be demonstrated between asteroids of S taxonomic type and the LL class of meteorites. JAXA launched the Hayabusa-2 mission on December 3, 2014, with the aim of returning a sample from the primitive (i.e., relatively unprocessed) C-class NEA (162173) Ryugu, which has a diameter of about 900 m. The launch of the Origins Spectral Interpretation Resource Identification and Security-Regolith Explorer (OSIRIS-REx) took place on September 8, 2016. The sample mechanism is designed to collect between 60 g and a few kilograms, depending on the surface properties of the target, namely the primitive B-type NEA (101955) Bennu, which has a diameter of ~450 m. The spacecraft will reach Bennu in 2018 and return a sample to Earth in 2023.

2.2 Astronomical observations It is important to understand the requirements for rapid acquisition of mitigation-relevant parameters of the threatening object when an emergency arises, and to have a good overview of the ranges of parameter values (shapes, rotation rates, albedo, taxonomy/composition, etc., and any size-dependence of these) present in the NEO population. Given that the NEO discovery rate is still increasing, with more and more smaller objects being found by search programs of increasing sensitivity, we need to increase efforts to investigate the physical properties of NEOs, especially potentially hazardous objects, that are relevant to predicting orbital drift (due to, e.g., the Yarkovsky effect), the design of deflection campaigns, and potential damage on the ground. A further motivation for increasing efforts in physical characterization is the identification of suitable, representative, NEOs for deflection test missions.

Radar is a powerful method for the characterization of NEOs, especially their sizes, shapes, and surface structure. A radar echo contains information not only on the position and velocity of a NEO, but also on a number of mitigation-relevant physical parameters. Radiation transmitted at a single frequency is returned from a rotating asteroid with a spread of (Doppler-shifted) frequencies, each component frequency being associated with a particular time delay depending


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on the distance to the reflecting surface element. The “delay-Doppler” distribution of echo power is determined by the size, spin rate, orientation, and shape of the target asteroid, and radar reflectivity of the surface material. The strength of the echo, normalized to the size and distance of the target (“radar albedo”), can provide information on the mineralogy of the asteroid surface, in particular its metal content. A circularly polarized transmission will give rise to a mix of “opposite sense” (OC) and “same sense” (SC) polarized radiation in the echo, depending on the number of reflections taking place at the surface. The ratio of SC/OC polarization in the echo is an indicator of the roughness of the surface at the scale of the radar wavelength (typically 3–13 cm). In particular, radar observations have observed a large number of binary NEAs, and have taught us that small asteroids can have very irregular shapes but also that some NEAs are surprisingly round, with equatorial bulges, suggestive of loose agglomerates of boulders and gravel that change shape as the spin rate exceeds a critical value, presumably preceding the spinning off of material that may then accumulate to form a moon.

Observations in the visible and infrared spectral regions of light reflected from asteroid surfaces are a powerful means of studying their compositions and assigning them to taxonomic classes. Efforts to derive information on the physical properties of asteroids by observing absorption features in reflected sunlight are complemented by observations of thermally emitted infrared radiation and radar investigations.

A survey of the sizes and albedos of more than 100,000 asteroids has been carried out by the NASA WISE space telescope. Following in the footsteps of the successful Infrared Astronomical Satellite (IRAS) mission in 1983, WISE was launched to Earth orbit in December 2009 carrying a 40-cm-diameter telescope and infrared detectors. WISE surveyed the sky for 12 months and the objects observed included a total of at least 584 NEOs, of which more than 130 were new discoveries (Mainzer et al., 2011b [RD27]). The specially funded Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE) program analyzed images collected by the WISE spacecraft to derive information on the NEOs detected. The fact that the cryogenic phase of the WISE mission measured asteroid thermal emission in up to four infrared bands, centered on 3.4, 4.6, 12, and 22 μm, allowed reliable values of diameter, albedo, and other parameters to be derived for many of the asteroids observed.

In addition to size and albedo information, application of thermal models to thermal-infrared observations has revealed their potential to provide information on thermal inertia and mineralogy. The thermal inertia of an asteroid’s surface provides a guide to its porosity and cohesion. Low values of thermal inertia are consistent with a dusty, porous regolith, such as that of the Moon, while high values presumably indicate a relatively dust-free surface dominated by rock or rubble of higher density and thermal conductivity. Knowledge of the surface properties of a NEO is obviously important for any deflection technique that requires contact, or interaction, with the surface. Work carried out in NEOShield-1 has shown that the application of the Near-Earth Asteroid Thermal Model (Harris, 1998 [RD18]) to WISE measurements can reveal the presence of significant amounts of metal on the surface of asteroids ([AD2], Harris and Drube, 2014 [RD16]). A threatening NEA containing a large amount of metal would presumably be relatively robust and massive, depending on its internal structure, factors that would require careful consideration by deflection-mission planners and/or those mandated to manage mitigation, e.g., evacuation and other activities on the ground in advance of a possible impact. Furthermore, the Yarkovsky effect, due to anisotropic thermal emission, induces drift in the orbits of small asteroids. The magnitude of the Yarkovsky effect depends on thermal conductivity and therefore thermal inertia, knowledge of which is important for impact probability predictions.


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2.3 Review of results from NEOShield-1 and relevant thermal physics Results obtained in NEOShield-1, D2.1, indicate a systematic association of relatively high values of the NEATM fitting parameter η (the so-called beaming parameter) with M-type asteroids and high radar albedos, suggesting that η is a potentially useful tracer of metal content in asteroids (Fig. 1). The association appears to hold for MBAs and NEOs. Our results also imply that there are many metal-rich asteroids, possibly also in the NEO population, that have not yet been identified as such. The proportion of metal in an object is important for mitigation-relevant considerations of object density and robustness, and for identifying asteroids that may offer valuable resources for future ventures in the field of planetary materials exploitation.

In simple thermal modeling the NEATM η parameter is a measure of the departure of an asteroid’s temperature distribution from that of an object with a smooth surface and zero thermal inertia, or zero spin, in thermal equilibrium with insolation, i.e. as described by the standard thermal model (STM, η=1). A rotating spherical asteroid with non-zero thermal inertia will have a cooler sub-solar region and a longitudinal temperature distribution on the “evening” side that does not fall to zero at the terminator, in contrast to the STM case. An exception arises if the spin vector is in the solar direction (solar aspect angle = 0, see Fig. 2), in which case the sunward hemisphere has an STM-type temperature distribution with a maximum at the pole facing the Sun, and η is independent of thermal inertia and spin rate; in fact the value of η will be less than unity in this case due to the “beaming effect” of a rough surface, which gives rise to a higher measured temperature than expected for a smooth surface due to enhanced sunward emission from surface elements on the sunward hemisphere that happen to be oriented towards the Sun. In theory, the effects of a rough surface act so as to reduce η, while those of thermal inertia and rotation cause η to increase.

As discussed by Spencer et al. (1989 [RD40]), η can be taken as a proxy for the dimensionless “thermal parameter”, Θ:

Θ = Γ ω0.5 / (ε σ TSS3), (1)

where Γ is the thermal inertia, ω the spin frequency, ε the emissivity, σ the Stefan Boltzmann constant, and TSS the subsolar temperature. Thermal inertia is defined as:

Γ = (κ ρ c)0.5 , (2)

where κ is the thermal conductivity, ρ the density, and c the specific heat. The thermal parameter, Θ, and η both increase with increasing thermal inertia and increasing spin rate. Values of η range from about 0.5 for an object with a rough surface and very low thermal inertia (and/or very low spin rate), in which case Θ ~ 0, to around 3.0 for a rapidly spinning object with very high thermal inertia, in which case Θ >> 1. In contrast to Θ, η also depends on the spin-axis orientation with respect to the solar direction, as discussed above. Spencer et al. (1989 [RD40]) define Θ as a property of the body itself by setting the temperature in eq. 1 equal to the subsolar equilibrium temperature:

Θ = Γ ω0.5 R3/2 / [(1-A)3/4 ε 1/4 σ1/4 S13/4], (3)

where R is the heliocentric distance in AU, A the bolometric Bond albedo, and S1 the solar constant at 1 AU. Since the Sun’s spectral energy distribution peaks in the visible region and the dependence of asteroid albedos on wavelength is normally small, it is usual to assume that A = AV = q pV, where q is the phase integral. This allows the physically significant parameter A to be linked directly to the observationally derived visible geometric albedo, pV. In the standard H, G magnitude system described by Bowell et al. (1989 [RD1]), in which H is the absolute magnitude and G is the slope parameter, q = 0.290 + 0.684 G.

It is clear from our NEOShield-1 results that relatively high values of η are associated with metal-rich M-type asteroids (Fig. 1). However, questions remain concerning the dependence of η on taxonomic class in general, whether low values of η in the case of many M-type asteroids are really due to low solar aspect angles or differences in mineralogy, and whether the distribution


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of η values can be used to provide rough quantitative information on the thermal inertia associated with taxonomic or other groups of asteroids.

Summary: The results of NEOShield-1 work suggest that η, a model parameter referred to as the beaming parameter, which can be taken as a proxy for the dimensionless “thermal parameter”, Θ, is a potentially useful tracer of metal content in asteroids. The association appears to hold for MBAs and NEOs. Our results also imply that there are many metal-rich asteroids, possibly also in the NEO population, that have not yet been identified as such.

Figure 1: a. WISE η values (from Masiero et al, 2011: [RD30]) versus infrared albedo for main-belt asteroids. Basic taxonomic types are shown as colored bullets. The C and S types are from the SMASS taxonomic system (Bus & Binzel 2002 [RD3]). The red curve is a plot of the mean of the highest 10 η values in bins of 100 data points; the purple curve is the same after removal of all the currently identified M types from the dataset (either originally classified as such on the Tholen (1984 [RD42]) system or classified as X on the Bus-DeMeo system and having pv in the range 0.075 - 0.3). Objects with default values of pIR in the WISE dataset have been excluded, as have objects with fractional uncertainties in η and pIR exceeding 20%. b. Radar albedo versus WISE near-infrared albedo, pIR, for main-belt asteroids. The radar data are from Magri et al. (2007 [RD25]) and Shepard et al. (2010, 2012 [RD37], [RD38] ). The broad clustering into 3 groups seen in Fig. 1a is also evident here, despite the relatively small number of points, whereby the central group here corresponds to high radar albedo, and in Fig. 1a to a peak in η and the location of the M types (from Harris and Drube, 2014 [RD16]).


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3 Further analysis of published infrared observations of asteroids Note: Part of the following work, including Fig. 2, a similar version of Fig. 3a, and Figs. 11, 12, 14, 15, 20, 21, and Table 1, has been published in Harris and Drube (2016 [RD17]). Open access version: https://arxiv.org/ftp/arxiv/papers/1608/1608.06839.pdf

3.1 The dependence of η on solar aspect angle As discussed above, if an asteroid’s spin axis is oriented close to the solar direction, the surface temperature distribution and η will be largely independent of thermal inertia and spin rate. A simple illustration of the dependence of the NEATM fitting parameter, η, on θ is given in Fig. 2. The diagram shows how, for a model asteroid, the temperature distribution on the surface of the asteroid varies with decreasing solar aspect angle, leading to a decrease in η, which reaches a minimum at θ = 0°. Therefore values of η derived from observational data should depend on the spin-axis orientation with respect to the solar direction.

Fitted values of η are now available for thousands of asteroids, thanks to the productivity of projects such as WISE/NEOWISE. However, spin vectors are available for a growing but still severely limited number of asteroids (Warner et al. 2009 [RD42]). Much effort was invested in gathering the dates, times and geometric data of WISE observations of those asteroids with known spin vectors. The dates and times of the WISE observations of a particular target were accessed via the NASA/IPAC Infrared Science Archive General Catalog Query Engine using a script especially written for the purpose (note: at a late stage in our NEOShield-2 work the WISE project uploaded the WISE dataset to the Planetary Data System: https://pds.nasa.gov/, from which the times of the observations are now readily accessible). The dates and times enabled ephemerides of the target to be generated for the times of the WISE observations via another purpose-written script which addressed the NASA/JPL Solar System Dynamics Horizons e-mail interface. Knowledge of the ephemerides, in particular heliocentric ecliptic coordinates, allowed the solar aspect angle, θ, to be calculated.

Measured η values from WISE (Masiero et al. 2011 [RD30]) plotted against θ (Fig. 3) confirm that η does indeed decrease with decreasing sin θ, as anticipated from simple modeling considerations (Fig. 2). We conclude, therefore, that η carries useful information on the surface temperature distribution.

Figure 2: Illustration of the effect of the solar aspect angle, θ, on the surface temperature distribution of a smooth-surface model asteroid. The solar aspect angle is the angle between the spin vector of the asteroid and the solar direction. The surface temperature distribution is governed by the rotation rate, the thermal inertia, and θ. See text for definitions of the symbols.


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Summary: We have confirmed that fitted η values for MBAs with known spin vectors decrease with decreasing sinθ (Fig. 3, cf. Fig. 2), and therefore that measured η values carry useful information on the surface temperature distribution.

Figure 3a: Confirmation that there is a general decrease in η with decreasing sinθ. Plotted values of η (Mainzer et al. 2016 [RD29]) are for MBAs with known spin vectors (outliers may be due to poor spin-axis determinations or η values). The horizontal line represents the median of the plotted η values. The red points and line trace the running weighted mean of the η values (in bins of 20 points). Note that no information on solar aspect angle is used in generating the η values published by the WISE project.

Figure 3b: As Fig. 3a but here showing the effect of plotting η against solar aspect angle in degrees. A maximum in the η values around 90° is evident, as expected from consideration of Fig. 2.


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3.2 The NEATM beaming parameter and thermal inertia Given that η can provide insight into the temperature distribution on the surface of an asteroid, which is influenced by thermal inertia, we investigated the extent to which η could be used to provide information on thermal inertia. Our initial analysis approach was to define a normalized “thermal parameter”, referred to here as Θnorm/Γ with reference to eq. 3, such that Θnorm/Γ ≡ (ω/ω0)0.5 (R/R0)3/2/[(1-A/A0)3/4], i.e. excluding thermal inertia. Thus for each asteroid for which the necessary information was available, the variables in eq. 3 were expressed in terms of those of a reference or “standard” main-belt asteroid with the following parameters: heliocentric distance, R0 = 2.7 AU, rotation period = 8 hr, albedo (pv) = 0.2, such that Θnorm/Γ = 1 for an asteroid with parameters equal to those of the reference object. For example, if an observed asteroid had a rotation period of 2 hr, but other parameters equal to those of the reference object, then its value of Θnorm/Γ would be 2; if, in addition, it had R=2.0 AU, then its Θnorm/Γ value would become 2 x (2.0/2.7)3/2. The advantage of this approach is that the dependence of η on Θnorm/Γ provides information on Γ, independent of other variables contributing to Θ. The dependence of η on Γ can be modelled using the parameters of the reference asteroid and NEATM applied to fluxes generated by means of a thermophysical model.

The thermophysical model used to generate comparison thermal-inertia curves applicable to the reference asteroid assumes spherical geometry and a smooth surface. The model asteroid is rotated until the surface temperature distribution becomes stable, following the numerical approach outlined by Spencer et al. (1989, [RD40]). The flux values at various thermal-infrared wavelengths that would be detected by an observer were calculated for a number of solar aspect angles, θ, between 0° and 90°, taking account of the observing geometry (assuming a solar phase angle of 25°, typical for main-belt asteroids). The exercise was repeated for a number of realistic values of thermal inertia, Γ. The NEATM was applied to determine best-fit values of η for each set of fluxes generated. The resulting model curves of η versus sinθ x Θnorm/Γ were fit with 3rd or 4th order polynomials to facilitate overlaying on plots of measured η versus sinθ x Θnorm/Γ for comparison purposes. In order to account for a realistic degree of beaming (see Section 2.3), the model curves of η versus sinθ x Θnorm/Γ have been shifted downwards to intersect the η axis at η0 = 0.8. While the degree of beaming in each case depends on the surface roughness of the asteroid in question, we find that η0 =0.8 appears to reflect an overall representative degree of beaming given the data sets examined to date (our treatment follows the δη reduction approach of Spencer et al., 1989 [RD40]).

WISE η values provided by Masiero et al. (2011, [RD30]) for main-belt asteroids classed as M and C type are plotted against sinθ x Θnorm/Γ in Figs. 4 and 5, respectively (note that to avoid confusion with the symbols sinθ x Θnorm/Γ is written as “sinθ x norm. TP/Γ” in Figs. 4 and 5). In Fig. 4 (M types) a clear trend of increasing η with increasing sinθ x Θnorm/Γ is evident: for values of the latter << 1, i.e. including objects for which the solar aspect angle is close to 0, values of η are less than 1, whereas for values of sinθ x Θnorm/Γ exceeding 1.3, η significantly exceeds 1 in all cases. Of the 3 model thermal-inertia curves overlaid on the plot, the upper two appear to represent the overall trend best, i.e. Γ=50-200 J m-2s-0.5K-1 (SI units). In the case of the C-type plot (Fig. 5) the trend is very different. While a slight increase in η with increasing sinθ x Θnorm/Γ is apparent, the trend is much less steep than in the case of the M types, suggesting that this population has a much lower representative thermal inertia; in general a value of Γ exceeding 10, but below 100, SI units appears to represent the C-type population reasonably well.

Note that due to the wide range of shapes and possible degrees of surface roughness of individual asteroids, and the generalized thermophysical model used, the thermal inertia curves plotted for comparison purposes can only serve as general indicators of thermal inertia for the populations of asteroids considered; the thermal inertia of any asteroid falling on or near a particular curve may differ greatly from the value represented by the curve. Nevertheless, a comparison of Fig. 4 with Fig. 5 for M- and C-type asteroids, respectively, reveals strong evidence for a dependence of thermal inertia on mineralogy.


15

Figures 4 (left) and 5: WISE η values (from Masiero et al, 2011 [RD30]) versus sinθ x Θnorm/Γ for M-type and C-type asteroids, respectively. The M-type asteroids are those classed as M type by Tholen (1984 [RD42]). Included in the M-type plot are two points for the asteroid 347 Pariana from IRAS data. The curves represent modelled η trends for the values of thermal inertia, Γ (Jm-2s-0.5K-1 = SI units), given in the labels. The model curves were generated by applying the NEATM to fluxes calculated using a thermophysical model. The numbers printed adjacent to each point are the asteroid designations. The plots are suggestive of very different values of thermal inertia being associated with the two taxonomic types.

Figure 6 (left): WISE η versus sinθ x Θnorm/Γ for Rivkin et al. (2000 [RD36]) W-class asteroids. Figure 7 (right): the same for Rivkin et al. (2000 [RD36]) M-class, and Shepard et al. (2015 [RD39]) Mm-class asteroids, respectively, the latter signifying a high radar albedo and a composition probably dominated by metal. Note that some objects appear in both plots due to the different class assignments of Rivkin et al. and Shepard et al.

Outlying points in the plots may simply be due to erroneous η values, or they may reveal important information on the objects concerned, such as taxonomic misclassification or inaccurate spin-vector data. For example, in Fig. 5 the η value of the C-type asteroid 1317 appears to be too high compared to the bulk of C-type asteroids. The taxonomic type of 1317 is given by Tholen (1984 [RD42]) as CX:, implying the C-type classification is uncertain. Our results suggest the thermal inertia of 1317 is much higher than that of a typical C-type asteroid.

It is informative to consider sub-populations of the main taxonomic classes. For instance, Rivkin (2000 [RD36]) noted that certain M-type asteroids showed evidence for a 3 μm feature in their reflection spectra, which is thought to be due to hydrated minerals. The presence of hydrated minerals is considered incompatible with the surface properties expected of disrupted cores of differentiated bodies, i.e. inconsistent with a metallic composition.


16

Figure 8: WISE η versus sinθ x Θnorm/Γ for SMASSII X-type asteroids. The broad distribution of the X-type points is indicative of different mineralogical compositions being present in the taxonomic class.

In Fig. 6 we plot the asteroids exhibiting a 3 μm feature, and classed by Rivkin et al. as “W” on the same scale as Figs. 4 and 5, while in Fig. 7 we plot the objects classed by Rivkin et al. as “M”, and those classed as “Mm” by Shepard et al. (2015 [RD39]) signifying a high radar albedo and a composition probably dominated by metal. Unfortunately, the plot of the Rivkin et al. (2000 [RD36]) W-class objects is rather inconclusive due to the lack of points in the range sinθ x Θnorm/Γ > 1, although in general the η values are slightly higher than those of the C-type asteroids in Fig. 5, suggesting somewhat higher thermal inertia. On the other hand, the M-types in Fig. 7 include 3 points with high η values indicative of significantly higher thermal inertia, as would be expected for objects with surfaces rich in metal. The Mm-class asteroid #347 in Fig. 7 appears to be an outlier, but note that the higher of the two IRAS η values for this object plotted in Fig. 4 is consistent with a relatively high value of thermal inertia. It should be noted that 4 of the W-class asteroids in Fig. 4 are also classed as Mm, indicating that the interpretation of their

mineralogies is complex and not well understood (see Shepard et al. (2015 [RD39]). Figure 8 shows a similar plot of X-type asteroids, classed as such on the basis of the Small Main-Belt Asteroid Spectroscopic Survey, Phase 2 (SMASSII, Bus and Binzel, 2002 [RD3]). In contrast to the M-types (Fig. 4) and the C-types (Fig. 5), the broad distribution of the X-type points is indicative of a number of different mineralogical compositions being present. A number of outliers (in particular #71) merit closer examination in order to check, for example, the reliability of the adopted spin vector or the WISE η value. Many X-types appear to cluster between the Γ = 10 and 50 Jm-2s-0.5K-1 curves, indicating similar thermal properties to the C-types in Fig. 5. However, in contrast to Fig. 5, a number of X-types have η values in the range 1.3 – 1.8, indicative of much higher “M-class” thermal inertia (cf. Figs. 4 and 5). Note that some objects appear in more than one plot due to their being assigned to different classes in the various classification systems considered here.


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Figure 9: η versus radar albedo for Tholen M- and/or SMASSI X-type asteroids. The plot excludes objects with values of sinθ x TPnorm/Γ < 0.5, where θ is the solar aspect angle. A correlation of η with radar albedo is apparent, supporting the idea that η can be used as a proxy for thermal parameter, which in turn is sensitive to metal content.

We find an apparent dependence of η value on radar albedo, once objects with low values of sinθ x Θnorm/Γ are excluded from the dataset (Fig. 9). The data in Fig. 9 provide support for the claim of Harris and Drube (2014 [RD16] that η can provide a useful indicator of asteroids containing large amounts of metal.

Further insight into the issue of large η values tracing metal content can be gained from a close inspection of Fig. 10 in which η is plotted against rotation period. The majority of M- and X-type asteroids appear on the left-hand side of the plot (period ≪ 10 h), i.e., they have relatively high spin rates (the median spin periods of the M-, X-, B+C-, and S-type asteroids plotted in Fig. 10 are 5.4, 7.4, 8.9, and 9.9 h, respectively). Since the thermal parameter is proportional to ω0.5 (eq. 3), η also increases with spin frequency. It has been known for many years that M-type asteroids have faster spin rates in general than C- or S-types of similar size (see, e.g., Lagerkvist et al., 1998 [RD23]). Metallic asteroids are thought to have been produced in catastrophic collisions that disrupted differentiated asteroids releasing their metallic cores. The high spin rates are presumably related to the large energies involved in such events, and the high densities of the bodies released. However, there is to date no generally accepted explanation.


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Figure 10: Plot of η versus rotation period for main-belt asteroids of the main taxonomic types (denoted by the labels). The η values are taken from the WISE catalog of Masiero et al. (2011 [RD30]) for objects with known taxonomic types and spin vectors. It is evident from this plot that many M- and X-type asteroids (colored blue) have relatively rapid spin rates. The mean η value of the dataset, 1.05, is represented by the dashed line.

Summary: We have used a thermophysical model to generate theoretical curves of η versus sinθ x Θnorm/Γ for a number of realistic values of thermal inertia, Γ. The advantage of our approach is that the dependence of η on sinθ x Θnorm/Γ provides information on thermal inertia, independent of other variables contributing to Θ. Comparison of measured values of η for different taxonomic types with the theoretical curves reveals differences in typical values of thermal inertia associated with the taxonomic types considered. For example, M types show a clear trend of increasing η with increasing sinθ x Θnorm/Γ, with values of thermal inertia in the range Γ=50-200 Jm-2 s-0.5 K-1 representing the overall trend best. In the case of the C-types a slight increase in η with increasing sinθ x Θnorm/Γ is apparent but the trend is much less steep than in the case of the M types, suggesting that the C-types have a much lower representative thermal inertia; in general a value of Γ exceeding 10, but below 100 J m-2 s-0.5 K-1 appears to represent the C-type population reasonably well. In contrast to the M- and C-types, the broad distribution of the X-type points is indicative of a number of different mineralogical compositions being present. We find an apparent dependence of η on radar albedo, once objects with low values of sinθ x Θnorm/Γ are excluded from the dataset. Our results provide support for the claim of Harris and Drube (2014 [RD16]) that η can provide a useful indicator of asteroids containing large amounts of metal. Furthermore, we have confirmed that the majority of M- and X-type asteroids have relatively high spin rates. Since the thermal parameter is proportional to ω0.5, η also increases with spin frequency, which appears to be a contributory factor in the sensitivity of η to metal-rich asteroids.


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3.3 A NEATM-based thermal-inertia estimator for near-Earth asteroids We have shown above how η values appear to be correlated with metal content and radar albedo, strongly supporting the idea that η carries useful information on asteroid thermal inertia. But is it possible to obtain quantitative estimates of thermal inertia from η values derived with the NEATM?

There is now a significant number of thermal inertia values for asteroids available in the literature. Delbo et al. (2015 [RD8]) provides a convenient table of published thermal inertia values for some 60 asteroids derived from thermophysical modeling, including NEOs, main-belt asteroids, and trans-Neptunian objects (TNOs). For most of the NEOs in Delbo et al’s table we have obtained η values, either directly from the literature or by calculating them by applying the NEATM to thermal-IR flux measurements taken from the literature or from the WISE data archive. Uncertainty estimates for derived values of η differ considerably, depending on the source. Some authors give formal statistical 1σ error bars while others quote larger, more conservative, error bars, which reflect experience of the repeatability of η values derived from different sets of measurements. We have taken a conservative approach and assigned error bars of ± 20% to η, except where those from the original source are larger. In each case the thermal parameter was calculated using eq. 3, and the solar aspect angle, θ, was derived from the ecliptic coordinates of the NEO’s spin vector and its heliocentric ecliptic coordinates at the time of the observations. A complication in the use of η to obtain physical information on asteroids is the fact that it shows a dependency on solar phase angle, α, with values obtained at α > 20° increasing roughly linearly with α. In order to remove this effect we have used the linear relation derived by Mainzer et al. (2011b [RD27]) to normalize the η values to a solar phase angle of 50°, which is near the center of the range at which WISE observations of NEOs are made. A plot of normalized η (assuming uncertainties of ± 20%) versus the thermal parameter multiplied by sinθ (Fig. 11) shows a very significant positive correlation, which has not been previously demonstrated. The data used in Fig. 11 are listed in Table 1.

The theoretical relationship between η and the thermal parameter, as given on the basis of a thermophysical model, is not linear. In Fig. 11 we have superimposed a red curve to illustrate the form of the theoretical dependence of η on Θ, which rises asymptotically to a maximum value as Θ tends to infinity (cf. Spencer et al., 1989, fig. 5 [RD40]; Lellouch et al., 2013, fig. 5 [RD24]). While the red curve represents a more complex relationship, it is evident from Fig. 11 that the weighted linear fit to the data (continuous line) may serve as a very good approximation, at least in the range 0.75 < Θsinθ < 3.5. Due to the shortage of reliable data and the large uncertainties, any formal analysis more sophisticated than a linear fit is not warranted at the present time.

We find the best weighted linear fit to the data of Fig. 11 is given by:

ηnorm = ηnorm,0 + b x Θsinθ, (4)

where ηnorm = η + (50°-α°) x 0.00963 (see Mainzer et al. (2011b [RD27]), ηnorm,0 = 0.74, b = 0.38, and Θ is the thermal inertia. Clearly, this relation can be inverted to provide estimates of thermal parameter, and therefore thermal inertia, given a measurement of η. Combining eqs. 3 and 4, we obtain the following expression for thermal inertia, Γ, as a function of η:

Γ = (ηnorm - ηnorm,0)((1-A)3/4 ε 1/4 σ1/4 S13/4)/(b sinθ ω1/2 R3/2), (5)

If SI units are used the units of the resulting Γ values will be J m-2 s-0.5 K-1. Since there is little evidence that η depends on α for α < 20°, for the purposes of normalization we take α = 20° for phases angles below 20°.


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Figure 11: Normalized η versus thermal parameter x sin θ for the NEOs in the compilation of Delbo et al. (2015 [RD8]), where Θ is the thermal parameter and θ the solar aspect angle. The data set used here includes only those objects for which robust η values could be obtained. The η values have been normalized to a solar phase angle of 50°, as explained in the text. The continuous thick line is a weighted linear best fit given by ηnorm = 0.74 + 0.38 x Θ sinθ; the dashed lines represent 1 σ deviations from the best fit. The red curve is indicative of the form of the theoretical dependence of η on thermal parameter (it is not a formal fit). Independent measurements of η for the same object are included as separate data points. The dataset used is given in Table 1 (the fractional uncertainties in Θ sinθ derive from those in ΓDelbo).

We have used eq. 5 to estimate thermal inertia values for the objects listed in table 2 of Delbo et al. (2015 [RD8]), for which reliable values of the required parameters were accessible from the literature or could be derived from on-line databases. We have excluded objects for which Θsinθ is outside the range 0.75 – 3.5 (see above). Figure 12 shows a comparison of the thermal inertia values estimated from eq. 5 with those derived by means of sophisticated thermophysical modeling by the authors cited by Delbo et al. The value of eq. 5 as an estimator of thermal inertia is immediately evident, especially given that Fig. 12 includes not only the near-Earth asteroids in Fig. 11 but also main-belt asteroids, a Centaur, and trans-Neptunian objects. In nearly all cases the η-based estimates agree within the error bars with the values derived from thermophysical modeling as listed by Delbo et al. Note that the plot covers nearly 4 orders of magnitude of thermal parameter.


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Figure 12: Plot of thermal inertia, Γ, estimated from values of η normalized to α = 50°, as explained in the text, versus Γ derived by means of sophisticated thermophysical modeling for objects in table 2 of Delbo et al. (2015 [RD8]). The data set excludes objects with thermal parameter outside the range 0.75 – 3.5 (see text). The error bars on the y-axis result from the assumption of uncertainties in η of ± 20%. There is remarkable agreement between the two sets of thermal inertia values over nearly 4 orders of magnitude in thermal inertia. As in Fig. 11 values of η for an object derived from independent sets of data, e.g. IRAS and WISE observations, are treated as separate values. Thus some objects are represented by two data points. The letter “W” or “I” appended to an object’s name indicates the η value was derived from WISE or IRAS data, respectively (see Table 1). The RMS fractional deviation, (ΓDelbo-Γest)/ ΓDelbo, is 40.1%.


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Table 1: Data used in Figs. 11 and 12

Name G R

(AU) Period (h)

pV α° sinθ η η err ηnorm ηnorm err Θsinθ Γ Γerr ΓTP ΓTP err Notes

NEOs 433_Eros 0.15 1.62 5.27 0.21 31.0 0.82 1.07 0.20 1.26 0.24 1.46 138 68 150 50 1

433_Eros 0.15 1.13 5.27 0.20 <20 0.99 1.05 0.11 1.34 0.14 1.02 229 103 150 50 2

1580_Betulia 0.15 1.14 6.13 0.11 53.0 0.97 1.09 0.22 1.06 0.21 1.10 136 92 180 50 3

1862_Apollo 0.15 1.1 3.06 0.26 35.3 0.97 1.15 0.23 1.29 0.26 1.20 168 79 140 100 2

25143_Itokawa 0.21 0.98 12.13 0.19 109.0 1.00 2.85 0.57 2.28 0.46 2.58 1094 324 700 200 4

33342_1998_WT24 0.15 1.01 3.70 0.56 60.4 0.88 1.86 0.38 1.76 0.36 1.40 381 135 200 100 5

33342_1998_WT24 0.15 0.99 3.70 0.56 79.3 0.96 1.25 0.25 0.97 0.19 1.45 81 70 200 100 5

99942_Apophis 0.24 1.04 30.56 0.30 60.4 0.99 1.82 0.37 1.72 0.35 1.56 982 351 600 300 6

101955_Bennu 0.15 1.13 4.30 0.05 63.4 1.00 1.55 0.03 1.42 0.03 2.23 246 103 310 70 7

162173_Ryugu -0.115 1.29 7.63 0.07 22.3 0.99 1.83 0.37 2.10 0.42 2.62 541 169 400 200 8

162173_Ryugu -0.115 1.2 7.63 0.07 52.6 0.97 1.63 0.15 1.61 0.15 2.30 392 146 400 200 9

175706_1996_FG3 -0.041 1.23 3.59 0.04 54.9 1.00 1.27 0.25 1.22 0.24 1.07 142 72 120 50 10

175706_1996_FG3 -0.041 1.38 3.59 0.04 <20 1.00 1.15 0.23 1.44 0.29 1.27 172 71 120 50 11

341843_2008_EV5 0.15 1.03 3.72 0.11 73.0 1.00 2.04 0.44 1.82 0.39 3.14 404 147 450 60 10

308635_2005_YU55 -0.13 0.99 19.31 0.06 34.0 1.00 1.08 0.22 1.23 0.25 1.68 442 226 575 225 12

308635_2005_YU55 -0.13 0.99 19.31 0.06 34.0 0.87 1.08 0.21 1.23 0.24 1.47 505 253 575 225 12

MBAs + others 16_Psyche_I 0.15 3.16 4.20 0.12 <20 0.49 0.86 0.07 1.15 0.10 2.16 62 35 125 40 13

22_Kalliope_I 0.21 3.07 4.15 0.15 <20 0.19 0.75 0.03 1.04 0.04 0.82 118 83 125 125 13

22_Kalliope_W 0.21 2.93 4.15 0.17 20.3 0.82 1.08 0.05 1.37 0.06 3.29 62 27 125 125 14

32_Pomona_I 0.15 2.81 9.45 0.23 21.2 0.98 0.98 0.12 1.26 0.15 1.40 67 33 70 50 13

32_Pomona_W 0.15 2.39 9.45 0.25 24.8 1.00 1.05 0.14 1.29 0.17 1.12 90 42 70 50 14

44_Nysa_I 0.15 2.44 6.42 0.47 24.0 1.00 1.04 0.06 1.29 0.07 2.60 67 31 120 40 13

87_Sylvia_I 0.15 3.51 5.18 0.04 <20 0.92 0.94 0.09 1.23 0.12 2.33 38 19 70 60 13

87_Sylvia_W 0.15 3.31 5.18 0.05 <20 0.90 0.86 0.01 1.15 0.01 2.10 35 20 70 60 14

107_Camilla_I 0.08 3.71 4.84 0.04 <20 0.90 0.97 0.09 1.26 0.11 0.91 37 18 25 10 13

107_Camilla_W 0.08 3.74 4.84 0.06 <20 0.98 0.99 0.10 1.28 0.13 1.01 35 17 25 10 14

110_Lydia_I 0.2 2.88 10.93 0.16 20.7 0.96 0.98 0.14 1.26 0.18 2.51 73 36 135 65 13

110_Lydia_W 0.2 2.9 10.93 0.17 20.3 0.94 0.97 0.08 1.26 0.10 2.48 73 36 135 65 14

115_Thyra_I 0.12 2.47 7.24 0.25 24.0 0.87 0.94 0.08 1.20 0.11 1.03 71 37 62 38 13

121_Hermione_W 0.15 3.29 5.55 0.08 <20 1.00 1.06 0.13 1.35 0.17 0.96 49 22 30 25 14

130_Elektra_I 0.15 3.53 5.22 0.07 <20 1.00 0.94 0.05 1.22 0.07 1.09 35 18 30 30 13

130_Elektra_W 0.15 3.11 5.22 0.09 <20 0.98 0.94 0.04 1.23 0.05 0.90 42 21 30 30 14

277_Elvira_I 0.15 2.63 29.69 0.20 22.5 1.00 1.10 0.19 1.37 0.23 2.57 159 70 250 150 13

277_Elvira_W 0.15 3.14 29.69 0.20 <20 1.00 0.92 0.01 1.21 0.01 3.35 91 47 250 150 14

283_Emma_I 0.15 2.88 6.90 0.03 <20 0.53 0.73 0.03 1.02 0.04 1.28 59 44 105 100 13

283_Emma_W 0.15 2.77 6.90 0.03 20.7 0.62 0.84 0.01 1.12 0.01 1.43 73 43 105 100 14

306_Unitas_I 0.15 2.17 8.74 0.19 27.9 0.91 1.01 0.14 1.22 0.17 2.33 97 49 180 80 13

306_Unitas_W 0.15 2.71 8.74 0.20 21.5 0.85 0.85 0.02 1.12 0.03 3.03 59 35 180 80 14

382_Dodona_I 0.15 2.56 4.11 0.13 23.3 0.86 1.09 0.17 1.34 0.21 1.79 70 31 80 65 13

382_Dodona_W 0.15 2.76 4.11 0.14 21.3 1.00 1.58 0.02 1.86 0.02 2.34 100 33 80 65 14

694_Ekard_I 0.15 1.84 5.93 0.04 33.4 0.85 0.95 0.11 1.11 0.13 1.30 89 53 120 20 13

720_Bohlinia_I 0.15 2.89 8.92 0.14 20.3 0.86 1.11 0.17 1.40 0.21 2.46 94 40 135 65 13

956_Elisa_W11 0.15 2.76 16.49 0.15 21.1 0.91 0.99 0.04 1.27 0.05 1.19 104 50 90 60 15

1173_Anchises_W11 0.03 4.93 11.61 0.03 <20 0.91 1.35 0.14 1.64 0.17 1.81 65 24 50 20 15

2060_Chiron 0.15 14.87 5.92 0.18 <20 0.91 0.91 0.10 1.20 0.13 1.39 4.3 2.3 5 5 16

10199_Chariklo 0.15 13.51 7.00 0.04 <20 0.91 1.12 0.08 1.41 0.10 3.40 8.2 3.5 16 14 16

90482_Orcus 0.15 47.8 10.47 0.24 <20 0.91 0.98 0.05 1.27 0.07 1.23 1.1 0.5 1 1 16

136108_Haumea 0.15 51.1 3.92 0.80 <20 0.91 0.95 0.33 1.24 0.43 0.82 0.5 0.4 0.3 0.2 16

208996_2003AZ84 0.15 45.4 6.78 0.11 <20 0.91 1.05 0.20 1.34 0.26 1.63 1.2 0.5 1.2 0.6 16


23

Notes to Table 1: The column headed “Γ” contains values of thermal inertia (J m-2s-0.5K-1) estimated from η normalized to α = 50°, as explained in the text. The column headed “ΓTP” contains values of thermal inertia derived by means of detailed thermophysical modeling (Delbo’ et al. 2015 [RD8]). The data set excludes objects with Θ sinθ outside the range 0.75 – 3.5 (see text), and those for which a reliable value of η was not available or could not be calculated from the available data. The errors on the Γ values result from the assumption of uncertainties in η of at least ± 20% (see text). Spin vectors are from the sources cited or the Asteroid Lightcurve Database (Warner et al. 2009 [RD43]).

Data sources for the NEOs: 1. Harris & Davies (1999 [RD15]); 2. Harris (1998 [RD18]); 3. Harris et al. (2005 [RD19]); 4. Müller et al. (2005 [RD34]), η calculation this work; 5. Harris et al. (2007 [RD20]), with spin vector from Busch et al. (2008 [RD4]), the observations were of the “evening” and “morning” sides of the object; 6. Müller et al. (2014 [RD32]), η calculation this work (NEATM fit less secure, based on flux measurements on Rayleigh-Jeans side of thermal continuum only); 7. Emery et al. (2014 [RD10]); 8. Hasegawa et al. (2008 [RD21]), η calculation this work; 9. Campins et al. (2009 [RD5]); 10. This work using data from the WISE cryogenic archive; 11. Wolters et al. (2011 [RD45]); 12. Müller et al. (2013 [RD33]), η calculation this work, the two entries are for two possible pole solutions: for the purposes of Fig. 3 the mean of the corresponding two values of Θ sinθ has been taken.

Data sources for the MBAs and Jupiter Trojans: 13. For objects with names appended by “I” the source of the data is the IRAS SIMPS Catalog (Tedesco et al. 2002 [RD41]), η values were derived in this work; 14. For objects with names appended by “W” the source of the data is the WISE catalog of Masiero et al. (2014 [RD31]); 15. For objects with names appended by “W11” the source of the data is the WISE catalog of Masiero et al. (2011 [RD30]).

Data source for the remaining objects: 16. Lellouch et al. (2013 [RD24]).

For objects with unknown pole directions (the last 7 objects in the table) θ = 65° (sinθ = 0.91) was assumed, based on the mean value of sinθ for the objects with known θ; for lower sinθ the resulting thermal inertia would be higher.

We caution that the due to the paucity of data and the large uncertainties, eq. 5 should only be considered a preliminary indicator of thermal inertia at this stage. It is “tuned” for near-Earth asteroids (Fig. 11); while it evidently can provide useful estimates for other types of object, its performance, e.g. in the case of main-belt objects, is difficult to judge on the basis of available data: eq. 5 predicts values for several main-belt asteroids plotted in Fig. 12 that are well below the values derived from thermophysical modeling. However, in most of these cases the values derived from thermophysical modeling have very large error bars. Furthermore, in the case of large main-belt asteroids analysis of WISE thermal-flux data requires corrections for saturation effects. We find that excluding WISE data for main-belt asteroids in those cases in which IRAS data are available leads to an improvement in the RMS fractional deviation in Fig. 12 from 40.1% to 37.5%.

We have also investigated the expected accuracy of eq. 5 by considering three sources of uncertainty:

1. The slope of the weighted best-fit relationship between η and solar phase angle, α, given by Mainzer et al. (2011b [RD27]) is 0.00963 ± 0.00015. We find that the quoted uncertainty has a negligible effect on the slope of the linear fit of Fig. 11, modifying the numerical constants in eq. 4 only in the third decimal place. We have investigated how the uncertainty in the Mainzer et al. (2011b [RD27]) relationship propagates through to the thermal inertia values based on eq. 5 given in Table 1: again the results reveal an insignificant RMS fractional error of 0.82%.

2. The slope of the weighted linear fit in Fig. 11 has 1 σ uncertainties shown in the plot as dashed lines; propagation through eq. 5 to the thermal inertia values in Table 1 results in a RMS fractional contribution to the error budget of 17%.

3. We assume a conservative overall uncertainty of 20% in measured η values, as explained above, which contributes 49.7% to the error budget. This contribution is by far the most significant.

Adding the above three independent uncertainty contributions in quadrature gives an overall uncertainty of 52.5% in thermal inertia values estimated via eq. 5, which is larger than the results from the comparison of thermal inertia values estimated via eq. 5 with those derived via


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detailed thermophysical modeling discussed above. The fact that the theoretical error budget is somewhat larger may be due to our assumption of a 20% uncertainty in η being over-conservative.

An important issue in the thermal modeling of asteroids is surface roughness. In Fig. 11 the relation between η and thermal parameter depends on surface roughness: in the case of a population of objects with relatively rough surfaces the intercept on the η axis will be lower. However, at high values of Θ sinθ roughness plays a reduced role in determining the longitudinal temperature distribution. In the limit of very high Θ sinθ all elements on the surface at the same latitude have the same temperature, regardless of roughness, and η reaches a maximum value independent of roughness. For an illustration of η dependence on thermal parameter for different levels of roughness see fig. 5 of Lellouch et al. (2013 [RD24]). A linear fit derived from a set of objects having different surface roughness characteristics to the objects plotted in Fig. 11 would presumably have a modified intercept and slope, with rougher/smoother surfaces giving rise to a slightly steeper/shallower slope. Implicit in the use of eq. 5 is the assumption that the surface roughness of the asteroid in question is compatible with the slope of the linear fit in Fig. 11. In any case, the error analysis above indicates that the effects of different degrees of roughness are adequately accounted for in the conservative 20% uncertainty assumed for measured values of η, at least for the set of objects in Table 1.

Another source of uncertainty in η is the effect of shape. For highly irregular objects the temperature and therefore η value observed depend on rotational phase. Brown (1985 [RD2]) showed that in cases of marked departure from sphericity, use of simple models can give rise to significant errors due to differences in temperature distributions between ellipsoids and spheres. The overall effect is an increase in thermal lightcurve amplitude above that expected from the variation in projected area due to rotation. In practice, however, the error can be minimized by taking measurements at several points on the lightcurve, which is often done in thermal-infrared observations of asteroids. For example, the WISE cryogenic survey made an average of 10 detections of a typical asteroid or comet spaced over ~ 36 h (Mainzer et al. 2011a [RD26]). Again, the error analysis above indicates that the combined effects of factors influencing η are adequately accounted for in the conservative 20% uncertainty assumed for measured values of η.

Of course, individual objects may have physical characteristics differing considerably from those of the population in general, or the combination of observing geometry, thermal inertia, and rotation rate may conspire to give a significant “morning/evening” effect, which would increase/decrease η (an extreme case appears to be the two entries for 1998 WT24 in Table 1; see Harris et al. 2007 [RD20]). In such cases results obtained from eq. 5 may be less accurate. However, occasional anomalous results will not affect the conclusions of this study, which is based on observations of hundreds of objects.

Further data on asteroid thermal properties will enable eq. 5 to be optimized and the scope of its usefulness to be better defined.

As mentioned in Section 3.1 the number of asteroids having reliably determined spin vectors is much smaller that the number for which fitted values of η are now available. It is interesting in this context to note that the rotation periods of asteroids have a broad but well-defined distribution that peaks around 6-10 h (see Pravec et al., 2002 [RD35], figs. 1 and 2). Furthermore, asteroid spin-axis orientations relative to the solar direction are not random, with relatively few objects having sin θ < 0.4 (see Pravec et al., 2002, fig. 6). In fact, the sin θ values of the asteroids listed in Table 1 have a mean near 0.91, corresponding to θ = 65°. While the non-random distribution of spin vectors may be partly due to observational bias (Pravec et al., 2002 [RD35]) the growing body of data suggests that the YORP effect plays a role for the smaller objects (D < 30 km) and for the larger ones a primordial distribution persists (Hanus et al, 2011 [RD13]).

In the absence of spin-vector data, to what accuracy can eq. 5 be used to estimate thermal inertia, assuming reasonable default values of spin period (e.g. 8 h) and sin θ (e.g. 0.91)?


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Figure 13a (left frame) and 13b (right frame): The relative accuracy of thermal inertia, Γ, estimated from η assuming default values of 8 h and 65° for the spin period, P, and solar aspect angle, θ, respectively (left frame). Unsurprisingly the accuracy improves when measured values of P and θ from Table 1 are substituted (right frame) but there is still agreement to within a factor of 2 in most cases. A sub-set of objects in Table 1 was chosen such that the objects pass the filter 0.75 < Θ sinθ < 3.5 (see text) with default values of 8h and 65° for P and θ, respectively (left frame) and the original values given in Table 1 (right frame). The RMS fractional deviation, (ΓDelbo-Γest)/ΓDelbo, decreases from 55.1% with the default values of P and θ, to 38.5% when the values listed in Table 1 are substituted.

In order to test eq. 5 with default values of spin period and sin θ, we chose a sub-set of the data in Table 1 such that the values of Θ sinθ lie in the linear range of Fig. 11, namely 0.75 – 3.5, regardless of whether the original values of spin period and sin θ are taken or the default values of 8 h and 0.91, respectively. A comparison of the performance of eq. 5 measured against the “ground truth” thermal-inertia values of Delbo et al. (2015 [RD8]) is shown in Fig. 13, in which the results using the default values are shown in the left-hand frame (Fig. 13a) and those using the values listed in Table 1 in the right-hand frame (Fig. 13b). It is evident that the points tighten significantly around the y=x line when measured values are substituted for the defaults, as expected, but the distribution in Fig. 13a is only moderately broader (the RMS fractional deviation increases from 38.5% to 55.1%). It appears from these results that eq. 5 may often provide useful information on thermal inertia even for objects for which spin vector data are not available.

Summary: We have shown that there is a very significant positive correlation between η and the thermal parameter multiplied by sinθ, which has not been previously demonstrated. The corresponding relation can be inverted to provide a means to estimate the thermal parameter, and therefore thermal inertia, given a measurement of η. Our thermal-inertia estimator was used to generate approximate thermal-inertia values for a set of 34 objects on which sophisticated thermophysical modeling has been performed, including NEOs, MBAs, a Centaur, and trans-Neptunian objects. The thermal-inertia estimator appears to perform well: the RMS fractional difference between the η-based estimates and the (more accurate) values derived from thermophysical modeling is only 40.0%.


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Figure 14: Estimated thermal inertia versus rotation period for MBAs. The NEATM-based thermal-inertia estimator (eq. 5) was used to estimate values of Γ from η values given in the WISE catalog of Masiero et al. (2011[RD30]) for objects with known spin vectors (black points, some 660 objects; note that the data set excludes objects with Θ sinθ outside the range 0.75 – 3.5, such as those with very low thermal inertia). There is a clear trend to higher values of thermal inertia for rotation period > 10 h. Error bars have been omitted for clarity. Uncertainties of ± 20% in the WISE η values result in a mean fractional uncertainty of ± 47% for the plotted thermal inertia values. The median diameter of the MBAs in the dataset is 24 km. Available thermal inertia values from detailed thermophysical modeling (Delbo’ et al. 2015 [RD8]) are superimposed for comparison (red points with error bars).

3.4 Application of the NEATM-based asteroid thermal-inertia estimator

3.4.1 Main-belt asteroids

We have carried out a number of investigations with various data sets to test the overall validity, scope, and usefulness of eq. 5. An interesting and unexpected result is an apparent dependence of thermal inertia on rotation period for main-belt asteroids (Fig. 14).

Available thermal inertia values from thermophysical modeling (Delbo’ et al. 2015 [RD8]) are overplotted in Fig. 14 (red points) on the estimated data from this work. While the thermophysically-modeled thermal inertia values taken on their own are insufficient in number and range of spin rate for any conclusion to be drawn on spin-rate-dependent thermal inertia, they do appear to be consistent with the trend apparent in the thermal inertia values estimated using eq. 5 (note that the latter data set excludes objects with Θ sinθ outside the range 0.75 – 3.5, such as those with very low thermal inertia).

In the interests of an independent check on the behaviour of our thermal-inertia estimator, we investigated the dependence of η on rotation period. The expected relation between η and rotation period was calculated using a smooth-surface thermophysical model based on spherical geometry for a constant thermal inertia of 75 J m-2 s-0.5 K-1 (cf. Fig. 14). The smooth-sphere thermophysical model, which is based on the work of Spencer et al. (1989 [RD40]), includes the effects of thermal inertia implicitly by determining the temperature of each surface element numerically via solution of the one-dimensional heat diffusion equation. The total observable thermal emission is calculated by summing the contributions from each surface element visible


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Figure 16: Dependence of η on rotation period. Open diamonds: non-binary NEAs with rotation periods <10 h. Double open diamonds: non-binary, slow-rotator NEAs with rotation periods >10 h. Filled symbols: binary NEAs. Filled squares: different observations of the same object: (5381) Sekhmet. The relatively large η-values of the slow rotators suggest higher values of thermal inertia for these objects. From Delbo et al (2011 [RD9]).

Figure 15: η versus rotation period. The red continuous curve represents the expected relation between η and rotation period on the basis of a smooth-surface thermophysical model based on spherical geometry for a constant thermal inertia of 75 J m-2 s-0.5 K-1 (cf. Fig. 14). The η values have been normalized to a solar phase angle of 50°, as explained in the text. The curve is normalized at a rotation period of 4 h to η =1.37, the median value in the range 3.0 – 5.0 h. The horizontal line represents the median of the plotted η values. As rotation period increases the η values remain relatively high, consistent with increasing thermal inertia.

to the observer. The model infrared fluxes thus generated were fit by the NEATM to provide model η values. As is evident in Fig. 15 the resulting model η curve (red), which is normalized to the median measured η value at rotation period = 4 h, does not fit the data for rotation period > 10 h and constant thermal inertia. As spin period increases the data points remain relatively high, while the model curve decreases as expected for increasing period (eq. 1). The widening gap between the η values and the model curve is consistent with increasing thermal inertia, as indicated by eq. 5 and evident in Fig. 14.


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Figure 17a: WISE η values versus pIR (Masiero et al., 2011 [RD30]) for main-belt asteroids of the main taxonomic types (denoted by the labels). The vertical lines drawn at pIR = 0.15 and pIR = 0.3 enclose the region in which Harris and Drube (2014 [RD16]) found a concentration of high-η, M-type asteroids. The distribution of the taxonomic types in this plot is very similar to that in Fig. 1a. Error bars have been excluded for the sake of clarity (see Fig. 10).

Figure 17b: A rather different distribution is evident in this plot of Γ versus pIR. Whereas many M- and X-types in the pIR = 0.15 – 0.3 range appear to have relatively large η values (Fig. 17a), these taxonomic types do not generally stand out as having exceptional values of thermal inertia. Error bars have been excluded for the sake of clarity (see Fig. 12).

An association of higher η values with slow rotators was noted by Delbo et al (2011 [RD9]) in the case of near-Earth asteroids (Fig. 16), who suggested that it is possibly related to the YORP effect, i.e. modification of the spin rate of an an irregularly shaped body via the reflection and thermal re-emission of solar radiation. YORP acts on small asteroids more effectively than on large asteroids, therefore it is questionable whether YORP could also explain the slow rotators with high thermal inertia in Fig. 14. Pravec et al. (2002 [RD35]) discuss a mechanism by which the asymmetric distribution of escaping ejecta from asteroids with diameters ~100 km may lead to a slowing of the spin rate with many impacts. However, it is not clear how a relatively large impact rate could lead to higher thermal inertia. We return to the question of thermal inertia and rotation rate in Section 3.4.3.

Harris and Drube (2014 [RD16]) demonstrated that large values of the NEATM η parameter appear to correlate with large values of radar albedo and M-type taxonomy. We have used eq. 5

to estimate values of thermal inertia, Γ, for a sample of objects with fitted WISE η values, taxonomic classifications, and known spin vectors. Objects with low solar aspect angles (θ < 30°) at the time of the WISE observations have been excluded, due to the potential for large errors in Γ outside the linear range of Fig. 11. Figure 17a is a plot of η versus IR albedo, pIR, for


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comparison with Fig. 1a. The vertical lines enclose the range pIR = 0.15 - 0.3, which Harris and Drube found to be associated with high-η, M-type asteroids. The plot in Fig. 17a shows a very similar distribution to that of Fig. 1a of Harris and Drube (reproduced in Fig. 1a above), with many M- and X-type objects present in the range bounded by the vertical lines. Figure 17b is a plot of Γ versus IR albedo, which shows that the estimated thermal inertia values of M- and X-type asteroids falling in the pIR = 0.15 – 0.3 “metallic” range, while significant, do not in general stand well above those of objects in the other taxonomic classes. Relevant statistical results for the data set used in Figs. 10 and 17 are summarized in Table 2. The thermal inertia of M types appears to be higher in general compared to that of the B+C group, but not compared to that of the S-types. It appears, therefore, that thermal inertia alone cannot explain the fact that many M- and X-type main-belt asteroids have relatively large η values.

As discussed in Section 3.1, insight into the issue of large η values tracing metal content can be gained from a close inspection of Figs. 10 and 17. The majority of M- and X-type asteroids appear on the left-hand side of Fig. 10 (period < 10 h), i.e., they have relatively high spin rates.

Table 2: Statistical results for the data set used in Figs. 10 and 17.

Mean σ Median Number in sample

Spin period [h]

M-type 8.3 7.3 5.4 19

X 8.7 4.3 7.4 39

B+C 13.0 13.1 8.9 87

S 14.9 20.7 9.9 140

Thermal inertia (eq. 5) [SI units]

M 97 26 92 19

X 88 36 83 39

B+C 89 39 80 87

S 107 45 101 140

Thermal parameter

M 2.25 0.51 2.22 19

X 1.91 0.70 1.69 39

B+C 1.68 0.46 1.64 87

S 1.81 0.48 1.75 140

Note: A few objects with very low solar aspect angles (sin θ < 0.3) have been excluded from the dataset used for Table 3 due to the possibility of large errors as a result of departure from the linear section of the plot in Fig. 11.

The above results explain and emphasize the usefulness of η values in identifying metal-rich asteroids. The combination of significant thermal inertia and relatively high spin rates leads to relatively large values of thermal parameter, and therefore η. We note that metal-rich objects that give rise to high radar albedos may not necessarily have large values of thermal inertia, as estimated via eq. 5, due to the effects of surface porosity and roughness (see Section 3.3).

Summary: Our thermal-inertia estimator has revealed an apparent dependence of thermal inertia on rotation period in the case of main-belt asteroids, in the sense of thermal inertia increasing as


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Figure 18: Plot of normalized η versus radar albedo for the NEOs for which relevant reliable data were accessible from the literature or on-line data archives. The η values have been normalized to a solar phase angle of 50°, as explained in the text. For 13 out of 16 objects in the dataset with known spin vectors, sin θ > 0.8; none of the 16 objects has sin θ < 0.5. In the case of 6 objects radar albedo uncertainties are available; the mean uncertainty is 33%. We have assumed the same value for all other radar albedo values available from http://echo.jpl.nasa.gov/~lance/asteroid_radar_properties/nea.radaralbedo.html Independent measurements of η for the same object are included as separate data points.

rotation period increases (we return to this result in Section 3.4.3). Use of the estimator to explore thermal-inertia values of asteroids of different taxonomic types suggests that in the available dataset potentially metal-rich objects have thermal-inertia values that in general do not greatly exceed those of other taxonomic types. It appears, therefore, that thermal inertia alone cannot explain the fact that many M- and X-type main-belt asteroids have relatively large η values. However, the majority of M- and X-type asteroids have relatively high spin rates, which enhance η. The fact that large η values are indicative of metal-rich objects appears to be due to a combination of thermal inertia and relative high rotation rates.

3.4.2 Near-Earth objects

Unfortunately the set of NEOs with taxonomic classifications, reliable η values from model fitting to thermal-IR fluxes, and known spin vectors is much smaller than in the case of MBAs. We have collected together relevant available data from the literature, with the help of reliable on-line data sources (e.g., WISE datasets, the NASA Planetary Data System, EARN) to examine the dependence of η and thermal inertia on other physical parameters. A plot of η versus radar

albedo is shown in Fig. 18. It is clear that low radar albedos (≤ 0.2) are associated with low values of ηα=50° (≤ 1.5), while most objects with radar albedos > 0.2 have ηα=50° >1.5. Comparison of Fig. 18 with Fig. 9 shows that the ηα=50°/radar albedo correlation in the case of NEOs appears to be weaker than that for MBAs. It should be noted, however, that the objects in the dataset used for Fig. 9 are M- and X-type MBAs only, which may be relatively metal rich, whereas those in the dataset used for Fig. 18 have not been selected on the basis of taxonomic type. In fact, the


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Figure 19: Plot of thermal inertia versus radar albedo for the NEOs for which relevant reliable data were accessible from the literature or on-line data archives. Thermal inertia values are those derived from thermophysical modeling (Delbo et al., 2015 [RD8]) where available, otherwise eq. 5 was used to provide estimates based on η. For other details see caption to Fig. 18. The triangles denote lower limits due to unknown pole directions; a solar aspect angle of 90° was assumed in these cases. If the spin vector is oriented otherwise (apart from 180°) the thermal inertia value will be higher.

only M types in Fig. 18 are 6178 and 29075, while 341843 is listed as “CX”. Given the paucity of available data and the large uncertainties, we feel a more detailed analysis of the dependence of η on radar albedo for NEOs is not warranted at present. We note in conclusion that a weighted linear fit to the data of Fig. 18 gives a Pearson’s correlation coefficient of r = 0.62, with a probability that the result arises at random of 0.27%.

We have investigated the dependence of thermal inertia on radar albedo taking thermal inertia values derived from thermophysical modeling (Delbo et al., 2015 [RD8]) where available, otherwise using eq. 5 to provide estimates based on η. The data are consistent with a trend of thermal inertia increasing with radar albedo (Fig. 19), but the correlation is weaker than that of Fig. 18.

We emphasize that the volume and quality of currently available relevant data are insufficient to allow firm conclusions to be drawn on the relationships between η, thermal inertia, radar albedo, and mineralogical composition in the case of NEOs. However, taking our results for MBAs (Section 3.1.1) and NEOs together, we find evidence of a dependence of η on metal content, which is stronger than the dependence of thermal inertia alone on metal content. It is evident that the relatively large spin rates of many M- and X-type asteroids lead to enhanced values of η. While metal content is an important factor in determining the thermal conductivity, and therefore thermal inertia, of asteroid surface material, surface porosity is also important in determining thermal properties.

In any case, the relationship between thermal inertia and spin rate appears to be a complex one, deserving of further study: In Fig. 20 thermal inertia is plotted against spin period for the NEOs in the dataset of Delbo’ et al. (2015 [RD8]). A similar distribution is evident to that based on η values found by Delbo et al. (2011 [RD9]), which is reproduced in Fig. 16, and in this work for thermal inertia values of MBAs derived using eq. 5 (Fig. 14). For further discussion of the dependence of thermal inertia on spin rate see Section 3.4.3.


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Figure 20: Thermal inertia (J m-2s-0.5K-1) versus rotation period for NEOs in the dataset of Delbo’ et al. (2015 [RD8]). As in the case of MBAs, slowly-rotating NEOs appear to be associated with higher values of thermal inertia (note: 54509 YORP, which has a very short rotation period of 0.20 h, is not shown in this plot).

Summary: In the case of NEOs a plot of η against radar albedo reveals a significant positive correlation. The correlation in the case of thermal inertia versus radar albedo is weaker. However, the volume and quality of currently available relevant data are insufficient to allow firm conclusions to be drawn on the relationships between η, thermal inertia, radar albedo, and mineralogical composition in the case of NEOs. We have investigated the dependence of (thermophysically modeled) thermal inertia on rotation period and find evidence of a strong dependence similar to that found in the case of MBAs in Section 3.4.1.

3.4.3 Thermal inertia and rotation rate

As is evident in Figs. 14, 15 and 20, there appears to be a significant trend to higher thermal inertia with decreasing rotation rate, which is apparent for both MBAs and NEOs. A similar trend may also hold for Centaurs/TNOs, although the correlation is not significant beyond the 1.6 σ level, according to Lellouch et al. (2013 [RD24]; see their fig. 8). As mentioned above, mechanisms that can modify the spin rates of asteroids include the YORP effect and collisions, but it is difficult to imagine a single effect that could lead to a spin dependence of thermal inertia operating on all types and sizes of asteroids from NEOs to TNOs. We suggest the explanation for the possibly universal trend of increasing thermal inertia with decreasing spin rate lies not in external influences, but rather in the different depths to which the thermal wave penetrates in otherwise similar asteroids rotating at different rates. The depth at which the amplitude of the diurnal thermal wave decays to 1/e of its surface value, known as the skin depth, is given by ds = (2κ/ρcω)0.5 = (2/ω)0.5 Γ/ρc (Wesselink, 1948 [RD44]; Spencer et al. 1989 [RD40]). Evidence is accumulating that the uppermost surface layer of an asteroid is


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Figure 21: Thermal inertia versus skin depth. The datasets are those used for Figs. 14 and 20. In the case of the MBAs Γ, from the NEATM-based thermal-inertia estimator, is plotted for bulk density values of 1000 kg m-3 (open blue circles) and 3000 kg m-3 (filled red circles), assuming c = 680 J kg-1K-1; error bars have been omitted for clarity (see caption to Fig. 14). The continuous lines represent the envelope of the data set for ρ = 3000 kg m-3. The dashed horizontal line at Γ = 2500 J m-2s-0.5K-1 represents the thermal inertia of solid rock. The NEO data of Fig. 20 (from thermophysical modeling) are superimposed (black points with error bars), taking c = 680 J kg-1 K-1 and ρ = 3000 kg m-3. Note that values of thermal inertia can be converted to thermal conductivity by substitution of the assumed values of ρ and c in the expression κ = Γ2/ρc; for reference, taking ρ = 2000 kg m-3, c = 680 J kg-1K-1, the values of thermal conductivity corresponding to dusty lunar-like regolith (~50 J m-2s-0.5K-1) and solid rock (~2500 J m-2s-0.5K-1) are 0.002 Wm-1K-1 and 5 Wm-1K-1, respectively.

very different in terms of porosity, thermal conductivity and density compared to sub-surface layers just a few tens of centimeters below. Data collected at millimeter and submillimeter wavelengths from the MIRO radiometer/spectrometer on-board the Rosetta spacecraft, indicate that the main-belt asteroid 21 Lutetia has a highly insulating surface layer around 1 - 3 cm thick, with Γ < 20 J m-2 s-0.5 K-1, on top of a “transition region” in which Γ increases to 60 - 120 J m-2 s-0.5

K-1 some 10 - 50 cm below the surface (Gulkis et al. 2012 [RD12]). It appears that the thermal conductivity and/or the material density, and therefore the thermal inertia, increase rapidly as depth increases. The lunar surface is thought to have a similar profile on the basis of data gathered during the Apollo missions, with a low-thermal-inertia porous surface layer 1 - 2 cm thick, covering a compacted sub-surface layer with some 50% higher density and an order of magnitude higher thermal conductivity at a depth of > 3 cm (Keihm and Langseth, 1975 [RD22]). These results imply that the rapid increase of density and thermal conductivity with depth can cause an increase of about a factor 4 or more in thermal inertia just a few centimeters below the surface. The temperature distribution on the surface is largely determined by the thermal conductivity


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and density of material above the skin depth. Figure 21 is a plot of Γ against skin depth for the MBA dataset used in Fig. 14 for bulk density values of ρ = 1000 and 3000 kg m-3, assuming c = 680 J kg-1 K-1 (Chesley et al. 2003 [RD7]; Čapek, & Vokrouhlický, 2004 [RD6]); the density would be expected to increase with depth from the lower value, which is representative of porous surface material. Superimposed on the plot of Fig. 21 are the Γ versus ds data for the NEOs of Fig. 20, for c = 680 J kg-1 K-1 and ρ = 3000 kg m-3. It is evident that the thermal inertia rises rapidly with skin depth, consistent with the findings for the Moon and Lutetia (see above), and the rise is steeper for NEOs than for MBAs. Extrapolation of the trends in Fig. 21 suggest that thermal inertia values representative of solid rock (~ 2500 J m-2s-0.5K-1, Golombek et al. 2003 [RD11] and references therein, for Martian rocks with diameter > 20 cm) are reached some tens of centimeters to meters below the surface in the case of the MBAs (the median diameter in our dataset = 24 km). In the case of the much smaller (km-sized) NEOs our results indicate that the porous surface layer is thinner, and suggest that large pieces of solid rock exist just a meter or less below the surface. Note that the thermal inertia values derived from observational data are effective values relating mainly to the material layers above the skin depth, ds. Assuming thermal inertia increases with depth, the effective values should be considered lower limits for the thermal inertia at depth = ds. Summary: The thermal inertia of asteroids, as determined by thermal-infrared observations, exhibits a strong dependence on spin rate, increasing as spin rate decreases. We interpret our discovery of spin-rate-dependent thermal inertia in terms of rapidly increasing material density and thermal conductivity with depth. Thermal inertia appears to increase by factors of 10 (MBAs) to 20 (NEOs) within a depth of just 10 cm. Our results are consistent with a very general picture of rapidly changing material properties in the topmost regolith layers of asteroids, as found in the case of 21 Lutetia (Gulkis et al. 2012 [RD12]). For a spin period > 10 h, knowledge of the rotation rate of an asteroid is crucial to choosing an appropriate value of thermal conductivity for calculation of the Yarkovsky effect; these results are also of relevance to modelers concerned with the mass and velocity distributions of ejecta expelled by a kinetic impactor spacecraft, and the corresponding ejecta-related momentum enhancement factor.

3.5 Further investigations with the WISE data

3.5.1 Using default values for the solar aspect angle

In Fig. 13 of Section 3.3 it is evident that substitution of default values of 8 h and sin θ = 0.91 (based on mean values for a large number of objects), for the spin period, P, and solar aspect angle, θ, respectively, does not greatly degrade the overall accuracy of the thermal inertia estimator (eq. 5) compared to the use of measured values of P and θ from Table 1: there is still agreement to within a factor of 2 in most cases. The WISE data base contains thousands of objects with measured rotation rates and fitted η values but without information on spin-axis orientation. We filtered the available WISE data for entries with fitted η values and known rotation periods and calculated thermal inertia values, Γ, using eq. 5, substituting sin θ = 0.91 for objects lacking a pole solution. The resulting data set increases the number of available thermal inertia estimates from around 650 in Fig. 14 to some 6000 for main-belt asteroids.

While the vast majority of objects in the enlarged dataset have no taxonomic classification, the number of objects with estimated thermal inertia values and taxonomic classifications is considerably larger, albeit based on the assumption that sin θ = 0.91 for the additional objects.

The results of the following investigations based on the enlarged dataset should not be considered as robust as those described above, which are based on the smaller set of objects with full spin vector solutions.


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Figure 22: Thermal inertia versus rotation period for M-, P-, and D-type main-belt asteroids. The data selection has been limited to relatively short periods (cf. Fig. 14) to minimize distortion of the distribution through the dependence of thermal inertia on rotation period evident in Fig. 14. The different taxonomic types appear to be associated with slightly different values of thermal inertia, with M types having a significantly higher mean value than the P or D types.

The results presented in Table 2 show that M types appear to be associated with slightly higher mean and median values of thermal inertia than B, C, and X types. The data shown in Fig. 22 indicate that the same is true for M types compared to P and D types. The higher metal content of M types is presumably responsible for the higher thermal inertia but the surface structure, in particular porosity and particle size, must also influence the measured thermal inertia. An asteroid may have a high metal content but if the topmost surface layer is dusty and highly porous it may suppress the overall thermal inertia.

A plot of Γ versus rotation period for the enlarged dataset is shown in Fig. 23. The dependence of thermal inertia on rotation period evident in Fig. 14 is more strongly apparent in Fig. 23. The skin depth was calculated as described in Section 3.4.3. The results obtained with the enlarged dataset (Fig. 24) confirm the trend of the data plotted in Fig. 21 suggesting that thermal inertia values representative of solid rock are reached a few meters below the surface in the case of main-belt asteroids. There is an interesting change in gradient of the envelope of the data set on the high thermal inertia side around skin depth = 0.5 - 1 cm, which is not evident in the sparser data set of Fig. 21 and may indicate a change from a topmost dusty layer to coarser layers of regolith in which the thermal inertia increases less rapidly with depth. The blue points with error bars are the NEO data set derived from thermophysical modeling plotted in Fig. 21. The green points with error bars are NEOs for which thermal inertia has been estimated using eq. 5, assuming sin θ = 0.91 (see above) for objects lacking a pole solution. The green points broadly follow the trend of the blue points, with a steeper slope than that of the main-belt asteroids, strengthening our conclusion that the porous regolith layer of km-sized NEOs is thinner than


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Figure 23: Estimated thermal inertia versus rotation period for some 6000 MBAs. The NEATM-based thermal-inertia estimator (eq. 5) was used to estimate values of Γ from η values given in the WISE catalog of Masiero et al. (2011[RD30]) for objects with known rotation rates. For objects lacking a pole solution sin θ = 0.91 was assumed (note that the data set excludes objects with Θ sinθ outside the range 0.75 – 3.5, such as those with very low thermal inertia). The trend to higher values of thermal inertia with increasing rotation period evident in Fig. 14 is much stronger in this plot. Error bars have been omitted for clarity. Uncertainties of ± 20% in the WISE η values result in a mean fractional uncertainty of ± 47% for the plotted thermal inertia values. The median diameter of the MBAs in the dataset is 8.2 km.

that of main-belt asteroids and material with thermal inertia consistent with that of large pieces of solid rock is present less than 1 m below the surface.

As mentioned above, the kinetic impactor deflection method is much more effective if the momentum enhancement parameter due to ejected mass, β, is significantly greater than unity. If large pieces of solid rock are present just tens of centimeters below the surface of a NEO, as our results suggest is the case in general, an impact of a high-speed projectile (spacecraft) into the surface of a NEO would be expected to eject much more mass than in the case of a surface consisting of porous dusty material to depths of several meters.

While a demonstration space mission is the only way to gain confidence in the performance of a kinetic impactor for hazardous NEO deflection, our results suggest that it may be more effective than has been assumed to date.


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Figure 24: Thermal inertia versus skin depth. The main-belt-asteroid data (orange points) are from the recently available PDS NEOWISE archive [RD29]. Thermophysically modelled values of Γ for NEOs are plotted in blue (data set of Fig. 20); values of Γ for NEOs estimated from eq. 5 are plotted in green. For the orange and green points sin θ = 0.91 has been assumed for objects lacking a pole solution - if sin θ is lower, Γ will be higher, so some orange and green points may reflect significant underestimates (cf. the green and blue points). The continuous lines represent the envelope of the MBA data set (orange points). The dashed horizontal line at Γ = 2500 J m-2s-0.5K-1 represents the thermal inertia of solid rock; in this plot the skin depth calculation assumes the same value. Note that values of thermal inertia can be converted to thermal conductivity by substitution of the assumed values of ρ and c in the expression κ = Γ2/ρc.

Summary: The results of investigations based on an enlarged dataset derived from the WISE catalog using a reasonable default value for the solar aspect angle where no pole solution is available, should not be considered as robust as those based on the smaller set of objects with full spin vector solutions. Nevertheless, our results indicate that the thermal inertia of M-type asteroids is generally higher than that of P and D types. The higher metal content of M types is presumably responsible for the higher thermal inertia. It is important to note, however, that if the topmost surface layer is dusty and highly porous it may suppress the overall thermal inertia, regardless of the mineralogical composition. In a plot of Γ versus rotation period for the enlarged dataset the dependence of thermal inertia on rotation period evident in Fig. 14 is more strongly apparent. The results obtained with the enlarged dataset follow the trend of the data plotted in Fig. 21 suggesting that thermal inertia values representative of solid rock are reached a few meters below the surface in the case of main-belt asteroids. There is an interesting change in gradient of the envelope of the data set on the high thermal inertia side around skin depth = 0.5 - 1 cm, which is not evident in the sparser data set of Fig. 21 and may indicate a transition from a topmost dusty layer to coarser layers of regolith in which the thermal inertia increases less rapidly with depth. The enlarged dataset strengthens our conclusion that the porous regolith layer of km-sized NEOs is thinner than that of the relatively large main-belt asteroids and material with thermal inertia consistent with that of large pieces of solid rock is present less than 1 m below the surface.


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3.5.2 Thermal inertia estimates of NEOShield-2 targets

A search of the WISE data archive for thermal infrared observations of the targets in the NEOShield-2 work package 10 observation program revealed 13 NEOShield-2 targets for which fitted η values are available. Relevant observational data were obtained from the NASA/IPAC Infrared Science Archive General Catalog Query Engine and the NASA/JPL Solar System Dynamics Horizons Web-Interface. While spin periods are available for all of these small NEO targets spin-axis orientations are available for only three of them, which were measured in the course of the NEOShield-2 work; we assumed sin θ = 1.0 for the rest (if sin θ is less than 1.0, i.e. the spin axis is not aligned perpendicular to the solar direction, the thermal inertia values will be higher, so the tabulated values in these cases should be considered lower limits). Estimates of thermal inertia based on eq. 5 are given in Table 3.

Table 3: Thermal inertia and skin depth for NEOShield-2 targets.

Name G R

(AU) Period

(h) D

(km) pV Tax. α° sinθ η η err ηnorm ηnorm err Γ(eq. 5) Γerr Skin depth

(cm) NEOs 4055_Magellan 0.15 1.98 7.47 2.78 0.33 V 29.8 1.0 1.383 0.147 1.578 0.168 155 60 0.80

7822 0.15 1.19 2.39 1.21 0.133 S 57 1.0 2.261 0.452 2.19 0.439 354 107 0.91

25916 0.15 1.43 4.60 5.68 0.262 - 45.2 0.44 1.111 0.222 1.157 0.231 230 128 0.82

40267 0.15 1.75 4.96 1.64 0.45 Sq 35.2 1.0 1.073 0.101 1.216 0.114 85 45 0.30

68278 0.15 1.29 4.23 0.942 0.087 - 51.6 1.0 1.282 0.256 1.266 0.253 152 73 0.52

85628 0.15 1.27 2.82 1.01 0.302 51.1 0.99 2.097 0.419 2.086 0.417 307 95 0.86

88263 0.15 2.13 10.34 4.876 0.059 - 28.3 1.0 1.300 0.260 1.509 0.302 187 73 1.00

163243 0.15 1.19 6.23 1.55 0.202 - 56.6 1.0 2.184 0.437 2.121 0.424 527 162 2.18

194268 0.15 1.514 6.80 1.24 0.055 X 40.8 1.0 0.927 0.185 1.016 0.203 87 63 0.38

243566 0.15 1.61 14.37 1.46 0.091 C, X 38.0 0.96 1.853 0.393 1.969 0.418 492 167 3.10

243566 0.15 1.61 2.31 1.46 0.091 C, X 38.0 0.96 1.853 0.393 1.969 0.418 198 67 0.50

398188_Agni 0.15 1.03 21.99 0.43 0.16 S 81.0 1.0 2.473 0.138 2.174 0.121 1300 400 11

410777 0.15 1.62 5.87 0.47 0.01 Xc 37.5 1.0 2.421 0.176 2.541 0.185 445 130 2.0

441987 0.15 1.06 4.98 0.23 0.071 Sv 72.8 1.0 1.949 0.142 1.729 0.126 420 150 1.75

Notes: η, D, and pv values from NEATM model fitting are taken from Mainzer et al. (2011b [RD27]), with the exception of 410777 (2009 FD) for which the values are from Mainzer et al. (2014 [RD28]). The uncertainties in the Γerr column result from the assumption of an uncertainty in η of ± 20%, or that from the WISE data archive given in the ηerr column, whichever is larger. The estimates of skin depth assume c = 680 J kg-1 K-1 and ρ = 3000 kg m-3. Taking a value for ρ of 1000 kg m-3, which may be more appropriate for the topmost porous material, would increase the values of skin depth by a factor of 3 (see Fig. 21). Note that the two entries for 243566 reflect the current availability of two spin-vector solutions for this asteroid.

The thermal inertia values cover a broad range but the results of Section 3.4 demonstrate that care must be exercised when comparing thermal inertia values with other physical properties, such as taxonomic type. Thermal inertia should be considered in combination with rotation rate and skin depth. Thermal-infrared observations of slowly rotating asteroids probe relatively large depths. If physical properties, such as density and thermal conductivity, increase with depth, derived thermal inertia values will be correspondingly higher. Therefore, thermal infrared observations of identical asteroids with different spin rates may give very different values of thermal inertia.

Summary: The thermal inertia values of 13 NEOShield-2 targets estimated from WISE thermal-infrared observations range from about 80 to 1300 J m-2 s-0.5 K-1. Our results imply that in this dataset the broad range of thermal inertia is mainly due to the different depths probed by thermal infrared observations of asteroids with different spin rates, rather than differences in thermal properties of material on their surfaces.


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4 Conclusions 1. Values of the so-called “beaming parameter” (η values), derived by fitting the NEATM model (Harris, 1998 [RD18]) thermal-flux distributions to observational data, appear to correlate with thermal inertia and radar albedo. 2. For a set of near-Earth objects with thermal inertia values available from thermophysical modeling (Delbo et al., 2015 [RD8]), and fitted η values available from the literature or via published observational data, a correlation of η with thermal parameter (Spencer et al., 1989 [RD40]) is apparent, which can be approximated by a simple linear fit over a broad range of thermal parameter. 3. The approximate linear dependence of η on thermal parameter can be inverted to provide an expression for thermal inertia as a function of η (“NEATM-based thermal-inertia estimator”). The thermal-inertia estimator provides values of thermal inertia that are generally within a factor of 2 of the values derived from detailed thermophysical modeling compiled by Delbo et al. (2015 [RD8]), over nearly 4 orders of magnitude in thermal inertia. 4. While M-type asteroids (and X-types with albedos in the M-type range) generally have substantial thermal inertia, this factor alone appears to be insufficient to explain the correlations between η and radar albedo and η and thermal parameter. Our results demonstrate that it is the combination of significant thermal inertia and relatively high rotation rates that gives rise to relatively large values of thermal parameter, and therefore η, in the case of most metal-rich asteroids. These results explain and emphasize the usefulness of η values in identifying metal-rich asteroids. 5. The thermal-inertia estimator has revealed an interesting and unexpected tendency for slowly-rotating main-belt asteroids to have higher thermal inertia. A similar trend is found for NEOs on the basis of thermophysical modeling. Our results indicating increasing thermal inertia with decreasing spin rate throughout the MBA and NEO populations are consistent with a very general picture of rapidly changing material properties in the topmost regolith layers. These results may influence the choice of an appropriate value of thermal conductivity for calculation of the Yarkovsky effect, e.g. in calculations of impact probabilities of potentially hazardous objects. If large pieces of solid rock are present just tens of centimetres below the surface of a NEO, as our results suggest is the case in general, an impact of a projectile into the surface of a NEO should eject much more mass than in the case of a surface consisting of porous dusty material to depths of several metres. While a demonstration space mission is the only way to test the performance of a kinetic impactor for hazardous NEO deflection, our results suggest that it may be more effective in general than has been assumed to date. 6. Our “NEATM-based thermal-inertia estimator” has the potential to facilitate further statistical investigations into the thermal physics of asteroids, especially NEOs, that could provide valuable insight into mitigation-relevant physical properties such as mineralogy, density, and surface structure.


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Acknowledgements This publication makes use of data products from WISE/NEOWISE, which is a project of the Jet Propulsion Laboratory/California Institute of Technology, funded by the Planetary Science Division of the National Aeronautics and Space Administration. We gratefully acknowledge the JPL Solar System Dynamics web service [https://ssd.jpl.nasa.gov/horizons.cgi], the EARN database [http://earn.dlr.de/], and the MinorPlanet.Info Asteroid Lightcurve Database [http://www.minorplanet.info/lightcurvedatabase.html], of which we have made extensive use.

https://ssd.jpl.nasa.gov/horizons.cgi

http://earn.dlr.de/

http://www.minorplanet.info/lightcurvedatabase.html


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5 Appendix Responses to the review comments of M. Delbo’ (CNRS)

We are very grateful to Marco Delbo’ for his review comments communicated on 23 July 2017 by means of a scanned handwritten document. We list below the main points raised by the reviewer and insert our responses beneath his points:

Reviewer point #1: I like this work a lot. The possibility to estimate the value of the thermal inertia (Γ) from the η value is a great means to obtain information about Γ for samples of asteroids containing a large number of bodies. The great advantage is that the approach presented here allows Γ to be derived from literature η without the need to model fluxes by means of a thermophysical model. However, in order to calculate ω [the rotation frequency of the asteroid], and sin θ [where θ is the solar aspect angle] one needs to know the asteroid spin state. If this information is available also a shape model (or at least a/b, b/c) is available. At this point one can use a thermophysical model to analyse the infrared data.

Authors’ response: We agree of course that detailed thermophysical modeling, wherever possible, is preferable to an approximate result given by eq. 5. However, we question whether the use of a very rough shape model with no further physical information would produce results significantly more accurate than those of eq. 5 (see p. 24). Performing a comparison of the two approaches would make for a very interesting future project.

Reviewer point #2: The thermal-inertia (Γ) estimator [eq. 5] explicitly includes a term ω-0.5 [where ω is the rotation frequency of the asteroid] so it is an obvious consequence of eq. 5 that Γ grows with rotation period. I understand that η should absorb this variation with period, e.g. one would expect η to decrease with increasing P, but the problem that eq. 5 includes P explicitly remains.

Authors’ response: This is a good point that we have given much thought to and motivated us to calculate the theoretical dependence of η on period shown in Fig. 15. It is evident from Fig. 15 that as rotation period increases the η values remain relatively high, whereas they would be expected, on the basis of a simple thermophysical model with spin-rate-independent thermal inertia, to decrease with period (cf. the red curve in Fig. 15). The behaviour evident in Fig. 15 is consistent with thermal inertia increasing with rotation period. Furthermore, the thermal inertia values of NEOs derived from detailed thermophysical modeling (i.e. NOT using eq. 5) also exhibit a positive correlation with rotation period (Fig. 20), which strengthens our confidence in the validity of eq. 5 and our claim that the measured thermal inertia of asteroids in general increases with increasing period. Whether η absorbs the “ω-0.5” increase of thermal inertia with period exactly (when thermal inertia is held constant with depth) is a good question, which could be explored in the future by means of detailed thermophysical and NEATM modelling.

Reviewer point #3: The finding that thermal inertia varies with depth makes total sense and it is a very important result. This said, I wonder how the hypothesis of having density constant with depth is accurate. [One would expect both density and thermal conductivity to increase with depth].

Authors’ response: See p. 33 where we state: “It appears that the thermal conductivity and/or the material density, and therefore the thermal inertia, increase rapidly as depth increases.” We agree that both density and thermal conductivity may well increase with depth.

Reviewer point #4: Fig. 15 - I am not sure I understand how you can get η < 1 by fitting the NEATM to the fluxes of a smooth-sphere TPM. But admittedly, it is intriguing that the values of η remain constant. Bias in the measurements of the rotation periods? How accurate are these P ⩾ 100 h?

Authors’ response: In the caption to Fig. 15 we state: “The curve is normalized at a rotation period of 4 h to η = 1.37, the median value in the range 3.0 – 5.0 h.” In effect we reduce the level of the curve by δη to force it to fit the measured η values at P = 4 h, thus copying the beaming


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effect due to roughness from the measured data into the theoretical η curve (our treatment follows the δη reduction approach of Spencer et al., 1989 [RD40] - see p. 14). It is quite likely that the uncertainties in the longer rotation periods are larger but it would seem improbable that the errors conspire to mimic the effect of increasing thermal inertia with increasing period as seen in Figs. 14 and 23. Furthermore, in both figures the effect is also evident for P < 100 h.

Reviewer point #4: Figure 4 and related text: This is a very cool result. Congratulations! It supports the idea that some M-types could have more metal than C-types in their regoliths. I read the document not as thoroughly as I wished, but I did not see a discussion about possible SIZE EFFECTS. We know that thermal inertia of asteroids is size dependent. How do the sizes of the M-type sample compare with the sizes of the C-type sample? Thermal inertia is also temperature and thus heliocentric distance (R) dependent. Are the Rs of the M-types similar to those of the C-types?

Authors’ response: Good question. The sizes and heliocentric distances of the M- and C-type samples used in Fig. 4 are compared in the table below. The difference in heliocentric distance is small; the difference in mean diameter is less than a factor 2 and therefore probably insignificant in terms of size-dependent thermal inertia (cf. figure 9 of Delbo et al. 2015 [RD8]).

Number Mean

diameter (σ) (km)

Median diameter

(km)

Mean heliocentric

distance, R (σ) (AU)

Median heliocentric

distance (AU)

C-type sample 52 133.3 (72.5) 132 3.01 (0.49) 3.08

M-type sample 20 79.5 (36.3) 72.5 2.77 (0.30) 2.85

In order to investigate the possible effect of size further, we plot below the thermal inertia estimated using eq. 5 for the data set used in Section 3.5.1, of M types and C types. The C-type sample has been cut at the large diameter end at D = 140 km, to force the median diameters of the C-type and M-type samples to be the same, namely 61 km. Furthermore, the data selection has been limited to relatively short periods to minimize distortion of the distribution through the dependence of thermal inertia on rotation period evident in Figs. 14 and 23. The plot shows that the difference in thermal inertia between the C-type and M-type samples persists, with the M types having significantly higher thermal inertia.