Thermal inertia of near-Earth asteroids and implications for the
  magnitude of the Yarkovsky effect
Microsoft Word - TI_NEA_070302RSingleSpaced.doc
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Thermal inertia of near-Earth asteroids and 
implications for the magnitude of the Yarkovsky effect 
Marco Delbo1,2, Aldo dell’Oro1, Alan W. Harris3, Stefano Mottola3, Michael 
Mueller3 
1INAF-Oss. Astron. di Torino, via Osservatorio 20, 10025 Pino Torinese (TO), Italy 
2Observatoire de la Côte d'Azur B.P. 4229, 06034 Nice Cedex 4, France 
3DLR Institute of Planetary Research, Rutherfordstrasse 2, 12489 Berlin, Germany 
Submitted to ICARUS: November 26, 2006  
Revised: March 2, 2007. Accepted: March 5, 2007. 
No. of manuscript pages: 31 
No. of figures: 6 
No. of tables: 1 
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Running head:  The thermal inertia of near-Earth asteroids 
Editorial correspondence to: 
Marco Delbo’ 
Laboratoire Cassiopée 
Observatoire de la Côte d’Azur 
BP 4229 06304 Nice, cedex 04 - France 
Tel.: +33 (0)4 9200 1944 
Fax.: +33 (0)4 9200 3121 
E-mail: delbo@obs-nice.fr 
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Abstract 
Thermal inertia determines the temperature distribution over the surface of an asteroid 
and therefore governs the magnitude the Yarkovsky effect. The latter causes gradual 
drifting of the orbits of km-sized asteroids and plays an important role in the delivery of 
near-Earth asteroids (NEAs) from the Main Belt and in the dynamical spreading of 
asteroid families. At present, very little is known about the thermal inertia of asteroids in 
the km size range. Here we show that the average thermal inertia of a sample of NEAs in 
the km size range is 200±40 J m-2 s-0.5 K-1. Furthermore, we identify a trend of increasing 
thermal inertia with decreasing asteroid diameter, D. This indicates that the dependence 
of the drift rate of the orbital semimajor axis on the size of asteroids due to the 
Yarkovsky effect is a more complex function than the generally adopted D-1 dependence, 
and that the size distribution of objects injected by Yarkovsky-driven orbital mobility into 
the NEA source regions is less skewed to smaller sizes than generally assumed. We 
discuss how this fact may help to explain the small difference in the slope of the size 
distribution of km-sized NEAs and main belt asteroids. 
Keywords:  
Asteroids; Near-Earth Objects; Infrared Observations; Photometry; Dynamics. 
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1 Introduction 
Observations of asteroids in the wavelength range of their thermal-infrared emission 
(>5 μm) have been used since the 1970s (Allen, 1970) to determine the sizes and the 
albedos of these bodies. In recent years, thanks to the advances in detector technology 
and the availability of 10-m class telescopes on the ground, thermal-infrared observations 
of asteroids have improved in sensitivity. Increased efforts have consequently been 
devoted to deriving the sizes and albedos of near-Earth asteroids (NEAs; for reviews see 
Harris and Lagerros, 2002; Delbo’ and Harris, 2002; Delbo’, 2004; Harris, 2006 and 
references therein), in order to better assess the impact hazard these bodies pose to our 
planet and to improve our understanding of their relation to main-belt asteroids and 
comets (see Stuart and Binzel, 2004; Morbidelli et al., 2002). Furthermore, improvements 
in spectral coverage and the possibility of easily obtaining spectrophotometric data 
through narrow-band filters in the range 5 – 20 μm have allowed information on the 
surface temperatures of asteroids to be obtained. The spectrum of the thermal-infrared 
radiation received from a body is related to the temperature distribution on that part of its 
surface visible to the observer. Several factors play a role in determining the temperature 
distribution on the surface of an asteroid, such as the heliocentric distance, albedo, 
obliquity of the spin vector, rotation rate, and a number of thermal properties of the 
surface such as its thermal inertia. 
Thermal inertia is a measure of the resistance of a material to temperature change. It 
is defined as cρκΓ = , where κ is the thermal conductivity, ρ the density and c the 
specific heat capacity. The thermal inertia of an asteroid depends on regolith particle size 
and depth, degree of compaction, and exposure of solid rocks and boulders within the top 
few centimeters of the subsurface (see e.g. Mellon et al., 2000). At the limit of zero 
thermal inertia (the most simple temperature distribution model for asteroids), a body 
with a smooth surface would display a temperature distribution which depends only on 
the solar incidence angle i, (on a sphere, i is also the angular distance of a point from the 
subsolar point): 
1/ 4cos , / 2
0, / 2
SST T i i
 (1) 
The subsolar temperature, TSS, is determined by equating the total energy absorbed by a 
surface element to that emitted in the thermal infrared, i.e.: 
=  (2) 
where A is the bolometric Bond albedo, S is the solar constant, r is the heliocentric 
distance of the body, ε is the infrared emissivity, σ is the Stefan-Boltzmann constant and 
η is the so-called “beaming parameter”, which is equal to one in the case that each point 
of the surface is in instantaneous thermal equilibrium with solar radiation. The surface 
temperature distribution that one obtain for η=1 on a spherical shape is that of the so-
called Standard Thermal Model (STM, Lebofsky and Spencer, 1989) that was widely 
used to derive diameters and albedos especially of main-belt asteroids (MBAs). In the 
more realistic case of a body with finite thermal inertia and rotating with a spin vector not 
pointing toward the sun, the temperature distribution is no longer symmetric with respect 
to the subsolar point: each surface element behaves like a capacitor or sink for the solar 
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energy such that the body’s diurnal temperature profile becomes more smoothed out in 
longitude (see Spencer et al., 1989; Delbo’ and Harris, 2002; Delbo’, 2004). The hottest 
temperatures during the day decrease, whereas those on the night-side do not drop to zero 
as in the idealistic case of zero thermal inertia, implying non-zero thermal-infrared 
emission from the dark side of the body. 
However, the effect of thermal inertia is coupled with the rotation rate of the body. An 
asteroid rotating slowly with a high thermal inertia displays a similar temperature 
distribution to one rotating more rapidly but with a lower thermal inertia. The degree to 
which the surface of an asteroid can respond to changes in insolation can be characterized 
by a single parameter: this is the so-called thermal parameter Θ (e.g. Spencer et al., 
1989), which combines rotation period, P, thermal inertia, Γ, and subsolar surface 
temperature, TSS, and consequently depends on the heliocentric distance of the body. The 
thermal parameter is given by: 
SST P
Θ = . (3) 
Note that objects with the same value of Θ, although with different P or Γ display the 
same diurnal temperature profile, provided they have the same shape and spin axis 
obliquity (the angle formed by the object spin vector and the direction to the Sun). In the 
case of non-zero thermal inertia, because the temperature distribution is no longer 
symmetric with respect to the direction to the Sun, the momentum carried off by the 
photons emitted in the thermal infrared has a component along the orbital velocity vector 
of the body, causing a decrease or increase of the asteroid orbital energy depending on 
whether the rotation sense of the body is prograde or retrograde. This phenomenon, 
known as the Yarkovsky effect, (see Bottke et al., 2002) causes a secular variation of the 
semimajor axis of the orbits of asteroids on a time scale of the order of 10-4 AU/Myr for a 
main-belt asteroid at 2.5 AU from the Sun with a diameter of 1 km. The Yarkovsky effect 
is responsible for the slow but continuous transport of small asteroids and meteoroids 
from the zone of their formation into chaotic resonance regions that can deliver them to 
near-Earth space (Bottke et al., 2002; Morbidelli and Vokrouhlický, 2003). The 
Yarkovsky effect is also important to explain the spreading of asteroid dynamical 
families (Bottke et al., 2001; Bottke et al., 2006; Nesvorný and Bottke, 2004). Moreover, 
the emission of thermal photons also produces a net torque that alters the spin vector of 
small bodies in two ways: it accelerates or decelerates the spin rate and also changes the 
direction of the spin axis. This mechanism was named by Rubincam (2000) as the 
Yarkovsky-O’Keefe-Radzievskii-Paddack effect, or YORP for short. 
Knowledge of the thermal inertia of asteroids is thus important for a number of 
reasons: (a) It can be used to infer the presence or absence of loose material on the 
surface: thermal inertia of fine dust is very low: ~30 J m-2 s-0.5 K-1 (Putzig et al., 2005); 
lunar regolith, a layer of fragmentary incoherent rocky debris covering the surface of the 
Moon, also has a low thermal inertia of about 50 J m-2 s-0.5 K-1 (Spencer et al., 1989). 
Coarse sand has a higher thermal inertia, i.e. about 400 J m-2 s-0.5 K-1 (Mellon et al., 2000; 
Christiansen et al., 2003), that of bare rock is larger than 2500 J m-2 s-0.5 K-1 (Jakosky, 
1986), whereas the thermal inertia of metal rich asteroidal fragments can be larger than 
12000  J m-2 s-0.5 K-1 (Farinella et al., 1998, Table 1). (b) Thermal inertia is the key 
thermophysical parameter that determines the temperature distribution over the surface of 
an asteroid and therefore governs the magnitude of the Yarkovsky and YORP effects 
(Capek and Vokrouhlický, 2004). (c) It allows a better determination of systematic errors 
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in diameters and albedos derived using simple thermal models, which make assumptions 
about the surface temperature distribution and/or neglect the thermal-infrared flux from 
the non-illuminated fraction of the body (see Spencer et al., 1989, Delbo’, 2004, Harris, 
2005). However, at present, very little is known about the thermal inertia of asteroids in 
general, especially in the case of bodies in the km size range. 
The thermal inertia of an asteroid can be derived by comparing measurements of its 
thermal-infrared emission to synthetic fluxes generated by means of a thermophysical 
model (TPM; Spencer, 1990; Lagerros, 1996; Emery et al., 1998; Delbo’, 2004), which is 
used to calculate the temperature distribution over the body’s surface as a function of a 
number of parameters, including the thermal inertia Γ. In these models, the asteroid shape 
is modeled as a mesh of planar facets. The temperature of each facet is determined by 
numerically solving the one-dimensional heat diffusion equation using assumed values of 
the thermal inertia, with the boundary condition given by the time-dependent solar energy 
absorbed at the surface of the facet (see Delbo’, 2004). This latter quantity is calculated 
from the heliocentric distance of the asteroid, the value assumed for the albedo, and the 
solar incident angle. Macroscopic surface roughness is usually modeled by adding 
hemispherical section craters of variable opening angle and variable surface density to 
each facet. Shadowing and multiple reflections of incident solar and thermally emitted 
radiation inside craters are taken into account as described by Spencer (1990), Emery et 
al. (1998), and Delbo’ (2004). Heat conduction is also accounted for within craters 
(Spencer et al., 1989; Spencer, 1990; Lagerros, 1996, Delbo’, 2004). Surface roughness 
can be adjusted by changing the opening angle of the craters, the density of the crater 
distribution, or a combination of both. However, Emery et al. (1998) have shown that if 
surface roughness is measured in terms of the mean surface slope, θ , according to the 
parameterization introduced by Hapke (1984), emission spectra are a function of θ  only 
and not of the crater opening angle and crater surface density. We recall here that 
tan tana d
θ θ θ θ= ∫  (4) 
where θ is the angle of a given facet from the horizontal, and a(θ) is the distribution of 
surface slopes. The total observable thermal emission is calculated by summing the 
contributions from each facet visible to the observer. Model parameters (e.g. Γ, A, θ ) are 
adjusted until the best agreement is obtained with the observational data, i.e. the least-
squares residual of the fit χ2 is minimized, thereby constraining the physical properties 
(albedo, size, macroscopic roughness, and thermal inertia) of the asteroid. 
To date, TPMs have been used to derive the thermal inertia of seven large MBAs 
(Müller, T. G. and Lagerros, 1998; Müller, T. G. and Blommaert, 2004; Mueller, M. et 
al., 2006b), and five NEAs (Harris et al., 2005; Müller, T. G. et al., 2005; Mueller, M. et 
al., 2006a, Harris et al., 2007); values derived lie between 5 and ~1000 J m-2 s-0.5 K-1, i.e. 
Γ varies by more than two orders of magnitude. The applicability of TPMs is limited to 
the few asteroids for which gross shape, rotation period, and spin axis orientation are 
known. Multi-epoch observations are also required for obtaining a robust estimation of 
the thermal properties of asteroids via TPM fit. 
There is, however, an extensive set of thermal-infrared observations of NEAs in the 
km size range for which no TPM fit is possible (e.g. Veeder et al., 1989; Harris, 1998; 
Harris et al., 1998; Harris and Davies, 1999; Delbo’ et al., 2003; Delbo’, 2004; Wolters 
et al., 2005).  In order to overcome this limitation, we have developed a statistical 
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inversion method, described in Section 2, enabling the determination of the average value 
of the thermal inertia of NEAs in the km-size range. Our approach is based on the fact 
that, even though shapes, rotation periods, and spin axis orientations are not known for 
every NEA, the distribution of these quantities for the whole population can be inferred 
from published data (La Spina et al., 2004; Hahn, 2006). 
In Section 3 we compare the result from our statistical inversion method with the 
values of the thermal inertias of asteroids determined by means of thermophysical 
models, and we identify a trend of increasing thermal inertia with decreasing asteroid 
diameter, D.  
In Section 4 we describe the implications of the trend of increasing thermal inertia 
with decreasing asteroid diameter, in particular for the size-dependence of the Yarkovsky 
effect and the size distribution of NEAs and MBAs. 
2 Determination of the mean thermal inertia of NEAs 
The large majority of asteroids for which we have thermal-infrared observations have 
been observed at a single epoch and/or information about their gross shape and pole 
orientation is not available, precluding the use of TPMs. In these cases simpler thermal 
models such as the near-Earth asteroid thermal model (NEATM; Harris, 1998) are used 
to derive the sizes and the albedos of these objects. The NEATM assumes that the object 
has a spherical shape, and its surface temperature distribution is described by Eq. (1) and 
Eq. (2). However, the parameter η is not kept constant, as in the case of the STM, but is 
adjusted in the fitting procedure to allow the model spectral energy distribution to match 
the observed data. In order to derive a robust estimate of the η-value the NEATM 
requires observations at different, ideally well-spaced, wavelengths in the thermal 
infrared. The parameter η can be seen as a measure of the departure of the asteroid 
temperature distribution from that of the STM and is a strong function of the surface 
thermal inertia (Spencer et al., 1989; Harris, 1998; Delbo’, 2004). However, η depends 
also on parameters such as the macroscopic surface roughness, θ , the rotation period, P, 
the bolometric Bond albedo, A, the thermal-infrared emissivity, ε, the heliocentric 
distance, r, the gross shape of the body, , the sub-solar latitude, θSS, the longitude, φSE, 
and the latitude, θSE, of the sub-Earth point (Delbo’, 2004). In general we can write that 
 η≡η(ε, A, r, Θ(Γ, P),θ , θSS, θSE, φSE, ). (5) 
These parameters are usually not known for the individual objects, but their distributions 
can be estimated (or reasonably assumed) for the entire population. Note that a set of θSS, 
φSE, and θSE , which depend on the ecliptic longitude λ0 and latitude β0 of the pole of the 
body, also defines the value of the solar phase angle, α.  
Delbo’ et al. (2003) noted that qualitative information about the average thermal 
properties of a sample of NEAs could be obtained from the distribution of the η-values of 
the sample as function of the phase angle, α. In particular, the absence of large η-values 
(e.g. η > 2) at small or moderate phase angles (e.g. ≤ 45o), and the fact that η tends to ~ 
0.8 for α approaching 0o, was interpreted in terms of the NEAs having low thermal 
inertias in general. In subsequent work (Delbo’, 2004) it was found that for a synthetic 
population of spherical asteroids with constant values of A, r, Γ, P, and θ , but with pole 
directions randomly oriented, the distribution of the points in the (α, η) plane is strongly 
dependent on Γ. By varying Γ until the distribution of the synthetic points in the (α, η) 
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plane matched the one derived from the observations, Delbo’ (2004) obtained a best-fit 
thermal inertia for the NEAs equal to ~500 J m-2 s-0.5 K-1. Harris (2005), using a similar 
method on a larger database of η-values and neglecting the effects of surface roughness 
(θ =0o), derived a best-fit thermal inertia of ~300 J m-2 s-0.5 K-1. 
Here we improve on the above-mentioned work by determining the mean thermal 
inertia of NEAs using a rigorous statistical inversion method, based on the comparison of 
the distributions of NEATM η-values from the current NEA database vs. α, with that of a 
synthetic population of asteroids generated through a TPM, using realistic distributions of 
the input parameters P, θSS, θSE, φSE, and A derived from the literature (see Table 1 with 
published η-values from Harris, 1998, Harris et al. (1998); Harris and Davies (1999); 
Delbo’ et al. (2003); Delbo’ (2004); and Wolters et al. (2005). La Spina et al. (2004) give 
the distribution of λ0 and β0 for NEAs, and Hahn (2006) that of NEA rotation rates). In 
the following section we describe our method in detail. 
2.1 Model parameter space 
As a first step we studied the dependence of η on the relevant parameters of Eq. 5. This 
was done by choosing typical parameter values and showing how small perturbations of 
the assumed values affect η. For the purpose of this analysis we assume A0=0.073, r0=1.2 
AU (as we will show below, these are the average values of A and r for the NEAs in our 
sample), 0θ =36
o (the value derived for 433 Eros; Domingue et al., 2002), 0=sphere, and 
Θ0=1.0. Note that Θ0=1.0 corresponds to a thermal inertia of ~200  J m-2 s-0.5 K-1 for 
surface temperatures typical of NEAs and P = 6 hours, a rotation period representative of 
asteroids with sizes between ~0.15 and 10 km (Pravec et al., 2002). We will show in 
section 2.3 that Γ = 200 J m-2 s-0.5 K-1 is the mean thermal inertia of NEAs. The 
illumination and observation geometry was varied such that θSS was uniformly distributed 
in the range between 0 and π/2 and θSE, φSE were varied in such a manner that the 
resulting sub-Earth vectors were uniformly distributed over the celestial sphere. The 
values of θSE, φSE were further subject to the constraint that the phase angle be ≤ 100o. 
Figure 1 shows the sensitivity of η to a change in the model parameters. In particular, for 
each value of θSS, θSE, and φSE, the variation of η due to a 1% change in each parameter is 
plotted. We have also calculated, for some fixed illumination and observation geometries 
(e.g. θSE = 0o, φSE = 45o and θSS = 0o), how the variations Δη scale with changes in the 
model parameters. We found that Δη is proportional to ΔA, Δ 0θ , ΔΘ, and Δr within a 
large range of variation (>100%) of each parameter from its nominal value. Because for 
common asteroidal material the thermal-infrared emissivity is thought to be relatively 
constant, it has been fixed for this study at ε = 0.9. It is appropriate for objects with 
surfaces that emit a substantial portion of their thermal-infrared radiation shortward of 8 
μm (Lim et al., 2005). Mustard and Hays (1997) have also shown that the reflectance 
spectra of fine-grained particulate materials, thought to be representative of planetary 
regoliths, have values around 0.1 and in general smaller than 0.2 in the region 8 – 24 μm. 
Because the reflectance, R, and the emissivity are related by Kirchhoff’s law (R=1-ε), the 
measurements cited above implies that ε = 0.9 is a reasonable estimate for the thermal-
infrared emissivity of NEA surfaces. Moreover, from Eq. 2 one can calculate that 
Δη≈1.6Δε for ε≈0.9 and η≈1.5 (the average η for the NEAs for which this parameter was 
derived from observations; see Table 1). This implies that variations of ε in the range 0.8 
– 1.0 cause changes of η that are within the typical uncertainty of ~20% in the estimation 
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of η from observations. Note that the value of Δη/η = 20%, where  Δη is the uncertainty 
in η, is based on the reproducibility of η for those objects for which more than one 
measurement is available from independent data sets. Moreover, 20% is also the mean 
value of Δη/η of the “Delbo’ Thermal Infrared Asteroid Diameters and Albedos” 
database at the NASA PDS (Delbo’, 2006). In this dataset for those observations where 
Δη is present, its value was formally calculated from the measurements of the asteroids’ 
thermal infrared fluxes. 
The vast majority of the observations in our sample was obtained at a phase angle 
smaller than 80o, and within this range, Fig. 1 shows that the largest variation of η caused 
by a 1% change of A (the bolometric Bond albedo) is approximately 0.1%. Because the 
mean value of A for our sample is A =0.073 and the standard deviation is 0.04 (see 
Table 1), the variation of η due to the distribution of the albedos is smaller than 5% and 
thus small compared to the typical uncertainty of Δη/η ~ 20%.  For this reason we have 
utilized a constant value of 0.073 for A in our statistical inversion method. 
Moreover, the variation of η due to a 1% change in the macroscopic surface 
roughness is strongly phase angle dependent, but in general smaller than 0.2% for α in 
the range 0-60o. This implies that even a ±100% change in θ  causes a variation of η 
within the typical 20% uncertainty. Note that a ±100% change in θ  corresponds to a 
large variation of the macroscopic roughness, ranging from that of a completely smooth 
surface to one oversaturated by hemispherical craters. For those observations carried out 
at α > 60o, η is more sensitive to variations of θ . For the reasons above we have treated 
θ as a free parameter in the inversion method and searched for the value that best fits the 
observational data. 
The sensitivity of η to changes of the objects’ heliocentric distances is such that a 1% 
change of r corresponds to a maximum 0.7% change of η. As calculated for the values in 
Table 1, the heliocentric distances in our sample have a mean value of 1.2 AU and a 
standard deviation of 0.1 AU (~8%). The corresponding variation of η is approximately 
6% and therefore small. We thus took a constant value of 1.2 AU for r in our statistical 
inversion method. Only in two cases, namely those of the 29-06-1998 observation of 
(433) Eros and for the 22-03-2002 observation of (6455) 1992 HE is the variation of η 
due to the deviation of the heliocentric distance from the nominal value of 1.2 AU 
slightly larger than the error bars.  
We note here that Eq. (5) implicitly contains the assumption that seasonal effects do 
not affect asteroid surface temperatures. However, when Θ ≠ 0, asteroid temperatures 
always depend on the previous thermal history of the surface. Since NEAs have in 
general large orbital eccentricities, these bodies experience large variations of insolation 
as a function of their orbital position, which may lead to a seasonal component of the 
variation of their surface temperatures and thus of the corresponding η-values. To 
demonstrate that our working hypothesis of Eq. (5) is valid (i.e. seasonal components are 
negligible), we calculated η for several synthetic asteroids with the same physical 
characteristics, but with different orbits, in order to explore the effect of different levels 
of insolation. Orbits were chosen with eccentricities in the range 0 to 0.8 but with a 
common perihelion distance rp. For different values of the asteroid thermal inertia in the 
range 200-5000 J m-2 s-0.5 K-1 and rp in the range 0.5 - 1.5 AU, we found variations of 
only a few percent in the η-values calculated at rp. This leads us to conclude that seasonal 
variations in the η-values are small and that Eq. (5) is valid. 
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In general NEAs have elongated shapes, which may cause their surface temperature 
distributions to differ from that of a spherical object with the same surface properties and 
illumination geometry. We studied the sensitivity of η to deviations from the spherical 
shape by calculating η-values of tri-axial ellipsoids, the semiaxes of which were varied in 
the ratio ( )1// aa  with 1 ≤ a  ≤ 6, assuming A0=0.073, r0=1.2 AU, 0θ =36o, Θ0=1.0, and 
for random orientations of the shape with respect to the Sun and the Earth. We found that 
the distribution of Δη is a function of a  (with values of Δη increasing with increasing a ), 
where Δη is the deviation of η from that calculated using a sphere under the same 
illumination and viewing geometry. However, the relative error on η, Δη/η, is always 
smaller than ±10% for a  ≤ 5 and α ≤ 45o. For α > 45o, the mean value of the relative 
error, 〈Δη/η〉, is smaller than +15% and its standard deviation, σΔη/η, is smaller than 5% 
for a  ≤ 5.  Because the maximum lightcurve amplitude of our model ellipsoid is 
L≈1.25log a  mag, Δη/η is smaller than 20% if L ≤ 0.873 mag. This condition is in 
general satisfied for the NEAs in Table 1, for which the average value of L is around 0.6 
mag. 
We expect that the contributions to Δη due to variations of the model parameters A, r, 
θ , and the ellipsoid axial ratio a, stack up randomly, since deviations of these parameters 
from their mean values are fully uncorrelated (e.g. there is no apparent reason that an 
NEA with an albedo higher than the average is also observed at an heliocentric distance 
higher than the its average value). We performed some numerical experiments in order to 
cross check this assumption and found that the value of Δη is in general a good proxy of 
[(∂η/∂AΔA)2 + (∂η/∂rΔr)2  (∂η/∂θ Δθ )]1/2. Adding the effect of non-spherical shapes 
increases the value of Δη, but never systematically at phase angles < ~60o. It is clear that 
ellipsoids are highly idealized shapes and larger contributions to Δη may be expected in 
the case of real NEAs. 
Figure 1 shows that the sensitivity of η to changes in the thermal parameter is very 
similar to the sensitivity to changes in r, with variations of η in general no larger than 
0.5% for a 1% change of Θ. However, while the value of r for the asteroids in Table 1 is 
rather constant around the mean value of 1.2 AU, Θ can range between 0.1 and 20 
considering that thermal inertia can be anywhere between 10 J m-2 s-0.5 K-1 (the thermal 
inertia of large main-belt asteroids) and 2500 J m-2 s-0.5 K-1 (that of bare rock). This 
implies that the scatter in the η-values that we observe in the NEAs of Table 1 is mainly a 
function of α and Θ, which depends on the thermal inertia. If we assume the thermal 
inertia to be roughly constant within the NEA population for a given size, its value can be 
inferred from the distribution of the measured η-values versus α. This is the idea on 
which our statistical inversion method is based. 
2.2 Model populations 
Our inversion method requires η to be computed for all members of a synthetic 
population of NEAs as a function of Γ. The calculation of η was performed by 
numerically generating thermal-infrared spectra by means of a TPM and fitting them with 
the NEATM. As discussed in the previous sections, the parameters A, ε, r, θ , and  
contribute little to the variation of η within the expected parameter ranges. Therefore, 
they have been kept fixed to their nominal values throughout the modeling process. In 
order to keep the amount of computing time required for the inversion within reasonable 
limits, the values of η have been computed only once for all possible combinations of the 
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remaining parameters, and the results have been stored in a four-dimensional look up 
table. The granularity of the look up table was chosen to be small enough to cause 
changes of η of about 0.1 between two consecutive parameter steps. 
For each value of Γ, we then generated a large number (30,000) of synthetic objects 
whose parameters have random values with distributions that have been chosen to 
provide a reasonable match to the observed population of NEAs. In particular:  
(i) the distributions of the angles θSS, θSE, and φSE were computed starting from the 
distributions of the spin-axis orientation (λ0, β0) from La Spina et al. (2004), the 
phase angle α, the heliocentric ecliptic latitude βH, and the geocentric ecliptic 
latitude βE of the asteroids at the time of the infrared observations (see Table 1 and 
Fig. 2);  
(ii) the distribution of the thermal parameter was calculated starting from the 
distribution of the NEA rotation periods (Hahn, 2006) and by using a constant 
value of Γ. 
In Fig. 3 three such populations are shown that correspond to the Γ values of 15 
(green), 200 (red), and 1000 (blue) J m-2 s-0.5 K-1, respectively. We have superimposed the 
η values for the NEAs in Table 1 on the synthetic data plot.  
2.3 Best-fit procedure 
Figure 3 gives a clear visual impression of the dependence of η on Γ. We therefore 
used a formal best-fit technique to estimate the value of Γ for which a synthetic 
population best fits the observed data, under the assumption that Γ is constant for all 
objects in the observed sample. The method that we used to compare the observed data 
with the bi-dimensional distributions of the synthetic points in the (α, η) plane is based 
on the two-dimensional Kolmogorov-Smirnov metric (K-S; Press et al., 1992). The 
distance D of the K-S metric is used as the goodness of fit estimator (Press et al., 1992). 
Our best-fit procedure consisted of finding the value of Γ that minimizes the K-S distance 
D.  From here on, we indicate this value with the symbol Γ*. Figure 4, where we have 
plotted the K-S distance D as a function of Γ, shows that the function D (Γ) has a 
minimum at Γ=200 J m-2 s-0.5 K-1, which is the value of thermal inertia that we take for 
Γ*.  
We expect Γ* to depend upon the assumed value for θ , the distributions of NEA 
rotation rates, and also on the spin-axis orientations that we have used to produce the 
distribution of the input parameters θSS, θSE, and φSE. Moreover, the value of Γ* must be 
affected by the errors in the measurements of the thermal infrared fluxes, i.e. by the errors 
on the η-values taken from the literature. In order to study the sensitivity of Γ* to changes 
applied to the nominal values of the input parameters, we first varied θ  in the range 
between 0° (perfectly smooth surface) and 58° (corresponding to the surface completely 
covered by hemispherical craters). Figure 4 shows the function D (Γ) for three different 
values of θ . It clearly demonstrates that the value of Γ* only weakly depends on θ and 
that a high degree of surface roughness produces a better fit to the observed data. 
We also investigated the sensitivity of Γ* to changes in the input distributions of 
asteroids’ spin-axis orientations and rotation rates. Figure 4 shows the function D (Γ) 
obtained by using random spin-axis orientations uniformly distributed over the sphere 
instead of the nominal distribution. In that case, the best-fit thermal inertia increases to 
- 12 - 
250 J m-2 s-0.5 K-1, and to 230 J m-2 s-0.5 K-1 if the distribution of the rotation rates are 
assumed to be uniformly distributed between 4 and 10 hours, a case which we believe to 
be very extreme. 
The sensitivity of Γ* to the errors affecting the η-values from Table 1 was studied by 
performing extended Monte Carlo simulations, in which we randomly varied the values 
of the η-values within their error bars (using normally-distributed random numbers), and 
for each simulation of noise-corrupted data we calculated the best-fit thermal inertia. The 
standard deviation of Γ* was found to be 40 J m-2 s-0.5 K-1.  
Of course, we expect that the distribution of the data points in Fig. 3 derives from a 
population with a range of thermal inertias, and further investigation is required to 
understand what the relations are between Γ* and the parameters defining the population, 
such as the mean value of Γ and the standard deviation of its distribution. In order to 
answer this question, we applied our inversion method on (α, η) points obtained from 
synthetic populations of NEAs with known distributions of thermal inertia. We used 
random values of Γ uniformly and normally distributed, varying both the mean value and 
the standard deviation of the populations. We found that our fitting procedure, based on 
the minimization of the K-S distance D, is capable of retrieving a good estimate of the 
mean value for Γ of the populations in all cases.  We conclude that the average value of 
the thermal inertia for km-sized NEAs is 200±40 J m-2 s-0.5 K-1, which is about four times 
that of the lunar soil and corresponds to a surface thermal conductivity of 0.0150.0100.027
−  W 
m-1 K-1 assuming that the surface material density and specific heat capacity are in the 
range 1500-3500 kg m-3 and 500-680 J kg-1 K-1, respectively  (Britt et al., 2002; Farinella 
et al., 1998). 
The value of Γ* that we have derived by means of the best fit procedure is less than 
10% of that expected for a bare-rock surface (Jakosky, 1986). This implies that the 
surfaces of NEAs have in general significant quantities of thermally-insulating regolith. 
However, Γ* is also about four times higher than the value that has been determined for 
the lunar soil and more than ten times higher than the thermal inertia typical of large 
main-belt asteroids. This effect may be due to the fact that the regolith present on NEA 
surfaces is less mature and/or less thick than that of the Moon and the largest MBAs. The 
higher NEA thermal inertia can also be explained in terms of a coarser regolith and the 
exposure of rocks and boulders on the surface of these bodies, as clearly shown in the 
high resolution images of (433) Eros and (25143) Itokawa obtained by the NEAR 
Shoemaker and the Hayabusa missions, respectively. 
A population of asteroids with constant Γ=200 J m-2 s-0.5 K-1 gives the best fit to the 
dataset. Figure 3 shows, however, that five points with η > 2 are clearly significantly 
higher than the majority, indicating that these objects presented unusually low color 
temperatures to the observer, possibly due to higher-than-average thermal inertia (see 
Delbo’ et al., 2003 and Delbo’, 2004). To gain insights into the width of the distribution 
of the thermal inertia of km-sized NEAs, we fitted the observed distribution of the data 
points with a synthetic population in which Γ was assumed to be uniformly distributed 
between 0 and ΓMAX. The best fit was obtained for ΓMAX ~ 600 J m-2 s-0.5 K-1. This 
suggests that the large majority of km-sized NEAs in our sample have thermal inertia 
below this value.  
The average value of the thermal inertia was derived for a sample of objects whose 
diameter distribution is shown in Fig. 5. We use here the radiometric diameters as derived 
by the NEATM. The mean diameter of the sample is 3 km, but if we remove the asteroid 
- 13 - 
433 Eros, the mean diameter value decreases to 2 km. 433 Eros is much larger than the 
average size of the sample (see Fig. 5). In fact the median value of the diameter 
distribution (including 433 Eros) is 1.8 km. We note that the distribution of log D (where 
D is the diameter measured in km) is well fitted by a Gaussian distribution with a central 
value of 1.7 km. The standard deviation of the best-fit Gaussian function is 0.31 (in log 
D). We can thus conclude that the average value of the thermal inertia is representative of 
NEAs in the size range 0.8 – 3.4 km.  
3 Size dependence of asteroid thermal inertia  
The mean thermal inertia for the sample of NEAs with published η-values is 
consistent with the values derived by means of TPMs for (433) Eros (Mueller, M. et al., 
2006a), (1580) Betulia (Harris et al., 2005), (25143) Itokawa (Mueller, M. et al., 2006a; 
Müller, T. G. et al., 2005), and (33342) 1998 WT24 (Harris et al., 2007) for which values 
around 150, 180, 350, 630, and 200 J m-2 s-0.5 K-1 have been obtained respectively. Note 
that in the case of (25143) Itokawa, Müller, T. G. et al. (2005) have obtained a thermal 
inertia value of 750 J m-2 s-0.5 K-1 combining thermal-infrared observations gathered at 
ESO in 2004 with those obtained by Delbo’ (2004) in 2001. On the other hand, from the 
latter dataset of observations and a series of further observations of (25143) Itokawa 
obtained at the NASA-IRTF 3 m telescope with MIRSI in 2004, Mueller, M. et al. 
(2006a) derived a thermal inertia of ~350 J m-2 s-0.5 K-1 or ~800 J m-2 s-0.5 K-1 depending 
on whether the size of the body was obtained from the TPM or was forced to the radar 
value of Ostro et al. (2004). In this work we have taken the mean value and the extreme 
values of 350, 750, and 800 J m-2 s-0.5 K-1 as our best estimate for the thermal inertia of 
Itokawa and its uncertainty. Müller T. G. et al. (2004) have also attempted at deriving the 
thermal inertia of the small (~0.28 km) NEA 2002 NY40. They obtained a value of 100 J 
m-2 s-0.5 K-1 in the case that the size of the object was derived from the TPM, or 1000 J m-
2 s-0.5 K-1 if the body’s size was forced to the value obtained from radar observations. 
However, it is important to note that that the thermal inertia of 2002 NY40 was derived by 
assuming an equator-on view and a spherical shape for this object. The value of the 
thermal inertia derived from the TPM is in general strongly dependent on the pole 
orientation of the body. For this reason we expect the value of Γ derived for 2002 NY40 
be less reliable than the values obtained for the other NEAs, for which the pole 
orientation derived from lightcurve inversion was adopted. 
From thermophysical modeling, Müller, T.G. and Lagerros (1998) derived the 
thermal inertias of a number of the largest MBAs, namely (1) Ceres, (2) Pallas, (3) Juno, 
(4) Vesta, and (532) Herculina, obtaining the values of 10, 10, 5, 25, and 15 Jm-2 s-0.5 K-1, 
respectively. Using the same approach, Müller, T.G. and Blommaert (2004) derived a 
thermal inertia of 15 J m-2 s-0.5 K-1 for (65) Cybele, and Mueller, M. et al. (2006b) 
obtained Γ~50 J m-2 s-0.5 K-1 for (21) Lutetia. From the published plots of the goodness of 
the TPM fit to the thermal-infrared data as a function of Γ it is possible to deduce that the 
relative uncertainties for the thermal inertias of these asteroids are around 50%.  
From the comparison of the values of Γ mentioned above, it is clear that there is an 
increase in the thermal inertia from that of large MBAs with diameters of several hundred 
km to that of much smaller km-sized NEAs, and that the values of Γ obtained for km-
sized NEAs are about one order of magnitude or more higher than the values derived for 
large MBAs, but still an order of magnitude lower than the thermal inertia of bare rock 
(~2500 J m-2 s-0.5 K-1; Jakosky, 1986). In order to highlight the behavior of the thermal 
- 14 - 
inertia of asteroids as a function of their size, we have plotted the mean value of thermal 
inertia for NEAs and the values of the thermal inertia derived by means of TPMs against 
object diameters in Fig. 6. Small open circles represent the literature values derived from 
the application of TPMs. The large open diamond is the result from this work. The axis 
on the right-hand side gives the asteroid surface thermal conductivity k as a function of 
size, on the basis of k=Γ2/(ρc), with constant surface density ρ = 2500 kg m-3 and specific 
heat capacity c = 600 J kg-1 K-1. These values are reasonable assumptions for asteroid 
surfaces (Britt et al., 2002; Farinella et al., 1998). For the asteroid 2002 NY40 a bar 
between 100 and 1000 J m-2 s-0.5 K-1 is drawn. The thermal conductivity has also been 
constrained in the cases of (6489) Golevka (Chesley et al., 2003) and for asteroids in the 
Karin cluster (Nesvorný and Bottke, 2004). The values of the thermal conductivities 
derived by these authors have been converted to values of Γ assuming ρ=2500 kg m-3 and 
c=600 J kg-1 K-1. Fig. 6 shows that the resulting limits, based on the measurements of the 
Yarkovsky effect on these bodies, are in general agreement with our results. 
Figure 6 reveals a convincing trend of increasing thermal inertia with decreasing 
asteroid diameter, D, confirming the intuitive view that large main-belt asteroids, over 
many hundreds of millions of years, have developed substantial insulating regolith layers, 
responsible for the low values of their surface thermal inertia. On the other hand, much 
smaller bodies, with shorter collisional lifetimes, presumably have less regolith, or less 
mature regolith, and therefore display a larger thermal inertia. Deriving a functional 
dependence of the thermal inertia as a function of the size of the body has important 
implications for improving the models of the orbital mobility of asteroids due to the 
Yarkovsky effect and to better quantify systematic errors in radiometric diameters and 
albedos of small bodies based on the use of thermal models that neglect the effects of 
heat conduction, such as the STM. The graph in Fig. 6 suggests that, to the first order, 
thermal inertia in this size range follows a power law. Expressing Γ as  
 0d D
ξ−Γ =  (6) 
(a linear relation in the log Γ – log D plot), a linear regression gives best-fit values of 
ξ=0.48±0.04 and d0=300±47, where D is km and Γ in S. I. units (J m-2 s-0.5 K-1), and the 
1σ uncertainty is based on the assumption that the errors on the thermal inertia and 
diameter values are normally distributed. (The values of Γ for 2002 NY40, 6489 Golevka 
and the Karin cluster asteroids were excluded from the linear regression analysis).  
However, the slope ξ of Eq. (6) may assume different values in different size ranges, 
since there are reasons to suspect that the surface properties of large asteroids may be 
different to those of smaller bodies: for example, Bottke et al. (2005) showed that 
asteroids with D > 100 km and most bodies with D > 50 km in size are likely to be 
primordial objects that have not suffered collisional disruption in the past 4 Gy. These 
large bodies have spent sufficient time in the asteroid belt to build a regolith such that 
they would display a low thermal inertia independent of size. In this case ξ should be 
about zero for D larger than about 50 km. In the same study it was shown that asteroids 
smaller than ~30 km are statistically the remnants of catastrophic collisional disruption of 
larger parent bodies, and the smaller the object, the fresher the surface. In this latter case 
one may intuitively expect that a dependence of the thermal inertia on the asteroid 
diameter would be more likely to occur, implying ξ > 0 for D < 30 km. For these reasons 
we tried to fit the data piecewise, separating the NEAs from the MBAs: a linear 
regression of Eq. (6) for the MBAs only of Fig. 6, gives best-fit values of ξ=0.49±0.27 
and d0=300±150 (Fig. 6, dotted-line) in good agreement with the trend obtained by fitting 
- 15 - 
the whole dataset of thermal inertias. However, we note that the accuracy of this fit is 
poor and that the value of ξ is strongly influenced by the thermal inertia of 21 Lutetia. On 
the other hand, a fit of Eq. (6) for near-Earth asteroids only, gives best-fit values of 
ξ=0.36±0.09 and d0=300±45 (Fig. 6, dashed-line) which corresponds to a shallower 
dependence of Γ on D for sizes up to  20 km.   
A further distinction in the thermal properties of MBAs compared to that of NEAs is 
given by the different mean heliocentric distances of the two classes of body, causing 
NEAs to have average temperatures ~200 K higher than those of MBAs. The thermal 
conductivity in the regolith is temperature dependent (Keihm, 1984), and so is thermal 
inertia. This temperature dependence of Γ may alter the slope ξ of Eq. (6) when both 
NEAs and main-belt asteroids are included in the fit. Under the assumption that heat is 
transported in the regolith mainly by radiative conduction between grains, the thermal 
conductivity is proportional to T3, with T being the temperature of the regolith grains 
(Kührt and Giese, 1989; Jakosky 1986). In this case 3 2TΓ ∝ and, from Eq. (2), 3 4r−Γ ∝ , 
where r is the heliocentric distance of the body. On the basis of this dependence of Γ with 
respect to r, we corrected the values of the thermal inertias of the asteroids of Fig. 6 to the 
mean heliocentric distance rref of 1.7 AU. Although the correction factors are in general 
smaller than the errors affecting the values of Γ, the thermal inertia values of NEAs (r < 
rref) are reduced, whereas those of MBAs (r > rref) are increased, yielding a smaller value 
of the slope ξ=0.37±0.04 and d0=230±30. 
Furthermore, the make up of NEA surfaces can be modified by processes such as 
close encounters with planets causing tidal disruption that do not affect asteroids in the 
Main Belt. Such processes might have been able to alter or strip off the regolith of some 
NEAs. Thus, while NEAs may be a good proxy for small main-belt asteroids, more 
observations are needed to confirm this point. 
It is clear that with the present small number of asteroids for which we have an 
estimate of the thermal inertia it is difficult to reveal possible variations of ξ with respect 
to the mean trend, ξ ~0.4, in different size ranges. Nevertheless, Fig. 6 shows a clear 
correlation of Γ with asteroid size and that asteroids in the 1 – 30 km size range have 
values of Γ in general larger than 100 J m-2 s-0.5 K-1. The fact that thermal inertia increases 
with decreasing size and that the value of Γ for km and multi-km sized asteroids is at 
least ten times larger than the value derived for the largest main-belt asteroids, has a 
number of important implications. First of all, radiometric diameters and albedos of 
asteroids derived by means of thermal models neglecting the effects of thermal inertia, 
such as the STM, are likely to be affected by increasing systematic errors with decreasing 
size. Spencer et al. (1989) have studied systematic biases in radiometric diameter 
determinations as a result of the effects of thermal inertia, rotation rate, pole orientation, 
and temperature. They concluded that the STM systematically underestimates the 
diameters of objects with non-negligible thermal inertia, while overestimating their 
albedos. Because we find that thermal inertia increases with decreasing asteroid diameter, 
it is likely that the systematic underestimation of asteroid diameters (and overestimation 
of asteroid albedos) obtained from the STM increases for decreasing asteroid size. 
Moreover, the absolute value and size dependence of the thermal inertia for asteroids 
with diameters smaller than about 10-20 km have crucial implications for the magnitude 
of the Yarkovsky effect, which is an important phenomenon that offers an explanation for 
the dispersion of asteroid dynamical families and the slow but steady injection of bodies 
- 16 - 
into the dynamical resonances that eventually transport them from the main belt to near-
Earth space. 
4 Implications for the magnitude of the Yarkovsky 
effect 
Current models of Yarkovsky-assisted delivery of NEAs from the Main Belt 
(Morbidelli and Vokrouhlický, 2003) and the spreading of asteroids families (Bottke et 
al., 2001; Nesvorný and Bottke, 2004), assume that thermal inertia is independent of 
object size. In this case, the theory of the Yarkovsky effect predicts that the orbital 
semimajor axis drift rate of an asteroid, /da dt , is proportional to D-1 (Bottke et al., 
2002). However, the mean value of the thermal inertia derived for NEAs and the inverse 
correlation of this thermophysical property with asteroid size, demonstrated in this work, 
give rise to a different magnitude of the Yarkovsky effect and a modified dependence of 
/da dt  on the object diameter. In order to derive this modification one can directly insert 
the function 0d D
ξ−Γ =  in the formulas given by Bottke et al. (2002) and Vokrouhlický 
(1999) to explicitly calculate /da dt  as a function of the relevant parameters.  
Here we discuss the case of MBAs with D < 10 km, assuming the linearized theory of 
the diurnal component of the Yarkovsky effect of Vokrouhlický (1999), which yields  
1 0.5
+Θ+ Θ
. (7) 
Because Θ is directly proportional to Γ, for Θ 1, 1 1da dt D− −∝ Γ and hence 
1da dt Dξ −∝ . We found that this condition holds in general for small asteroids in the 
Main Belt: in fact for objects with D smaller than ~10 km Fig. 6 shows that the thermal 
inertia is in general >100 Jm-2s-0.5K-1, and at heliocentric distances >2 AU the surface 
temperatures of these bodies are in general smaller than 250 K (see e.g. Delbo’, 2004). 
Pravec et al. (2002) have also shown that asteroids with sizes between ~0.15 and 10 km 
have a typical rotation rate around 6 hours. Inserting these values in Eq. (3), we find that 
Θ is in general larger than ~2 for main-belt asteroids in this size range. By taking ξ ~ 0.4, 
as derived in the previous section for the Γ= d0D-ξ relation, we obtain that the size 
dependence of /da dt  due to the diurnal component of the Yarkovsky effect is 
proportional to D-0.6 rather than proportional to D-1, as generally assumed, which would 
hold true for a thermal inertia independent of asteroid size.  
We caution, however, that there are currently no reliable estimates of thermal inertia 
available for any object smaller than Itokawa (D ~ 350 m), so the relation derived for 
0.6/da dt D−∝ should be assumed to hold for objects with diameters in the range between 
~0.35 and ~10 km. Moreover, for asteroids smaller than 350 m and/or higher values of 
the thermal inertia, the seasonal component of the Yarkovsky effect may become 
significant and contribute to the average value of da/dt. From this analysis we conclude 
that, in the Main Belt, the drift rate in semimajor axis due to the diurnal component of the 
Yarkovsky effect increases with decreasing asteroid size more slowly than is normally 
assumed in models of the origin of NEAs and the spreading of asteroids families.  
The shallower dependence of the Yarkovsky effect on the diameter of the bodies 
caused by the inverse correlation of Γ with D has the important implication that the size 
distribution of the asteroids injected into the NEA source regions is less skewed to small 
- 17 - 
objects than generally assumed. In the following we briefly discuss some of the 
consequences of this: there is general consensus that the large majority of NEAs originate 
from the Main Belt via well defined “feeding zones” of dynamical instability (Morbidelli 
et al., 2002). Asteroidal material can gradually drift towards these NEA source regions as 
a result of Yarkovsky-driven semimajor axis mobility (Morbidelli et al., 2002; Morbidelli 
and Vokrouhlický, 2003). The cumulative size distribution of a population of asteroids in 
a given diameter range (e.g. 0.35 < D < 10 km) can be approximated by a simple 
exponential function of the form 
 0( )N D N D
α−> = , (8) 
Therefore, according to this asteroid delivery model, the difference in the exponent α 
between the bodies injected into the NEA source regions and the remaining population of 
asteroids in the Main Belt is of the order of ~1 if the semimajor axis mobility is 
proportional to D-1. The same difference in the value of the exponents holds for the NEA 
and the MBA populations in a comparable size range, assuming that the large majority of 
NEAs come from the Main Belt. 
The results of the latest studies that have analyzed the size distributions of NEAs and 
km-sized MBAs imply that this difference is closer to 0.5-0.7, in favor of a Yarkovsky 
effect less effective for smaller asteroids, which is consistent with the results of this work. 
We recall that Eq. (8) can be converted into a cumulative absolute visual magnitude H 
distribution of a population of asteroids with the form 
 0( ) 10
HN H N β′< = , (9) 
where the exponential slope of the absolute magnitude distribution, β, can be converted 
into the power-law slope of the diameter distribution via α = 5β (see, e.g., Stuart and 
Binzel, 2004) . Several authors (Rabinowitz et al., 2000; Bottke et al., 2000; Stuart and 
Binzel, 2004) agree that β  is in the range 0.35 – 0.39 for the NEA population, which 
implies a value of αNEA between 1.75 and 1.95. The size distribution of km- and sub km-
sized MBAs is less constrained than that of NEAs, since the known population is still 
rather incomplete for H  > 14 – 15 (corresponding to values of D between 6 and 3 km for 
a geometric visible albedo of 0.11), so that beyond this threshold only extrapolations of 
the known distribution can be made. Morbidelli and Vokrouhlický (2003) used the slopes 
derived by Ivezic et al. (2001) from the Sloan Digital Sky Survey (SDSS), namely, 
β=0.61 for 13 < H < 15.5 and β=0.25 for 15.5 < H < 18, to extrapolate the observed H 
cumulative distribution (as given by the Astorb catalog) to km sized asteroids, and use it 
in their model of the Yarkovsky-driven origin of near-Earth asteroids. Assuming β=0.25 
for 15.5 < H < 18 this would give a value of αMBA=1.25 for the slope of the cumulative 
size distribution of MBAs in the range 1 < D < 3 km. This value is in good agreement 
with the even slightly shallower size distribution (αMBA~1.2) of km and sub km sized 
MBAs (for 0.5 < D < 1 km) found by the SMBAS survey (Sub-km Main-Belt Asteroid 
Survey) obtained by Yoshida et al., (2003). Taking the values of α from the studies 
above, we find that αNEA - αMBA = 0.5 – 0.7, in good agreement with a drifting population 
of asteroids with 0.6/da dt D−∝ . 
However, Morbidelli and Vokrouhlický (2003) have also shown that the collisional 
re-orientation of asteroid spin axes (which resets the drift speed due to the Yarkovsky 
effect), the collisional disruption of the bodies during their slow drift towards the NEA 
source regions, and the YORP effect, tend to decrease the difference between αNEA and 
- 18 - 
αMBA. The addition of these phenomena along with the revised dependence 
of 0.6/da dt D−∝ due to the Yarkovsky effect may help to explain the even steeper size 
distribution of small MBAs, and thus a difference between αNEA and αMBA smaller than 
0.5 – 0.7 implied by the recent results of the Sub-Kilometer Asteroid Diameter Survey 
(SKADS; Davis et al., 2006), which found βMBA = 0.38, corresponding to αMBA = 1.9, for 
13 < H < 17. 
The value of the thermal inertia also plays an important role in the YORP effect 
(Rubincam, 2000; Vokrouhlický and Capek, 2002), which is a torque produced by the 
thermal radiation emitted by asteroids with irregular shapes causing a slow spin-up/spin-
down and a change of the spin axis obliquities of these bodies. In contrast to the 
Yarkovsky effect, YORP also acts on bodies with zero surface thermal conductivity. 
However, in the case of a thermal inertia significantly larger than zero (in contrast to the 
case of zero-conductivity), YORP preferentially drives obliquity toward two asymptotic 
states perpendicular to the orbital plane, and asymptotically decelerates and accelerates 
rotation rate in about an equal number of cases (Capek and Vokrouhlický, 2005). Capek 
and Vokrouhlický (2005) have shown that the acceleration of the rotation rate, /d dtω , is 
largely independent of the thermal inertia, whereas its value significantly affects the rate 
of change of the obliquity, /SSd dtθ , in the sense that the higher the thermal inertia the 
larger the mean value of /SSd dtθ . Capek and Vokrouhlický (2005) found the median 
value of the distribution of /SSd dtθ  for populations of Gaussian spheres increased from 
3.33 deg/My for Γ=0 J m-2 s-0.5 K-1 to 5.94 deg/My for Γ=39 J m-2 s-0.5 K-1 and to 8.60 
deg/My in the case of Γ=122 J m-2 s-0.5 K-1, which is a suitable value for bodies of about 5 
km according to our Fig. 6. Because our Γ (D) relation predicts an even larger value of 
surface thermal inertia for asteroids of 1 km in diameter, the YORP reorientation of the 
spin vector of asteroids becomes a more effective mechanism in the case of km-sized 
asteroids, capable of driving the rotation axis to the asymptotic state perpendicular to the 
orbital plane in just a few tens of millions of years.  
5 Conclusions 
The thermal inertia of an asteroid can be derived by comparing measurements of its 
thermal-infrared flux, at wavelengths typically between 5 and 20 μm, to synthetic fluxes 
generated by means of a thermophysical model (TPM). To date TPMs have been used to 
derive the thermal inertia of seven large MBAs and five NEAs. Although an extensive set 
of thermal-infrared observations of NEAs exists, application of TPMs is limited to the 
few asteroids for which the gross shape, the rotation period, and the spin axis orientation 
are known.  
In order to overcome this limitation, we have developed a statistical method enabling 
the determination of the thermal inertia of a sample of objects for which such information 
is not available. This method has been applied to a sample of NEAs with diameters 
generally between 0.8 km and 3.4 km. The resulting value, Γ = 200 ± 40 J m-2 s-0.5 K-1, 
corresponds to a surface thermal conductivity of about 0.03 W m-1 K-1. 
This value of thermal inertia and those derived by means of TPMs reveal a significant 
trend of increasing thermal inertia with decreasing asteroid diameter, D. Assuming that Γ 
is proportional to D-ξ we derive a best-fit value for the exponent of ξ ~ 0.4. 
- 19 - 
The dependence Γ(D) has important implications for the magnitude of the Yarkovsky 
effect. On the basis of our results, the size dependence of the orbital semimajor axis drift 
rate /da dt  of MBAs for ~0.35 < D < ~10 due to the diurnal component of the 
Yarkovsky effect is proportional to D-0.6, rather than the generally assumed D-1 
dependence for  size-independent thermal inertia. 
The modified dependence, 0.6/da dt D−∝ , implies that the size distribution of the 
objects injected by Yarkovsky-driven orbital mobility into the NEA source regions is less 
skewed to smaller sizes than generally assumed. This may help to explain the smaller-
than-one difference in the value of the exponents of the cumulative size distribution of 
NEAs and MBAs. 
We stress that the dataset on which our results are based is small and more multi-
wavelength, multi-epoch thermal-infrared observations of asteroids with known spin 
states are required to refine our conclusions on the size dependence of thermal inertia and 
its consequences. 
6 Acknowledgments 
We wish to thank Bill Bottke and David Vokrouhlický for earlier suggestions and 
comments that inspired us to develop the original concept of this work further, and the 
referees of the present paper, Stephen Wolters and Bill Bottke, for suggestions that led to 
significant improvements in the presentation. M. D. wishes to acknowledge fruitful 
discussions with A. Morbidelli and A. Cellino. 
- 20 - 
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Table 1 
D α  P  spin Asteroid 
(km) 
pV  A η r 
(AU) (°) (hrs) 
Obs. Date 
axis 
433 Eros           23.60 0.200 0.079 1.05 1.135 10 5.27 17-01-1975 Y a 
433 Eros           23.60 0.210 0.082 1.07 1.619 31 5.27 29-06-1998 Y a 
1580 Betulia        3.82 0.110 0.043 1.09 1.199 53 6.138 22-06-2002 Y b 
1862 Apollo         1.40 0.260 0.102 1.15 1.063 35 3.065 26-11-1980 - c 
1866 Sisyphus       8.90 0.140 0.055 1.14 1.609 35 2.4 29-06-1998 - d 
1980 Tezcatlipoca  6.60 0.150 0.059 1.64 1.129 63 7.252 31-08-1997 Y a 
2100 Ra-Shalom     2.79 0.080 0.031 2.32 1.174 39 19.8 21-08-2000 Y e 
2100 Ra-Shalom     2.50 0.130 0.051 1.80 1.195 41 19.8 30-08-1997 Y f 
3200 Phaethon       5.10 0.110 0.043 1.60 1.132 48 3.604 20-12-1984 - c 
3554 Amun           2.10 0.170 0.067 1.20 1.243 16 2.53 15-03-1986 - c 
3671 Dionysus       1.50 0.160 0.063 3.10 1.126 58 2.705 02-06-1997 - a 
5381 Sekhmet        1.50 0.220 0.086 1.90 1.213 44 3 22-06-2003 - d 
5381 Sekhmet        1.40 0.240 0.094 1.50 1.119 35 3 14-05-2003 - d 
5587 1990 SB        4.00 0.250 0.098 1.10 1.399 19 5.052 09-04-2001 Y d 
5587 1990 SB        3.57 0.320 0.126 0.84 1.210 42 5.052 10-05-2001 Y e 
6455 1992 HE        3.43 0.280 0.110 0.80 1.641 22      - 22-03-2002 - g 
6455 1992 HE        3.55 0.240 0.094 0.70 1.364 29      - 29-09-2002 - g 
9856 1991 EE        1.00 0.300 0.118 1.15 1.093 36 3.045 11-09-1991 - f 
14402 1991 DB       0.60 0.140 0.055 1.04 1.025 36 2.266 16-04-2000 - e 
19356 1997 GH3      0.91 0.340 0.133 0.98 1.406 5 6.714 11-05-2001 - e 
25330 1999 KV4      2.55 0.080 0.031 1.06 1.392 3 4.919 14-05-2003 - d 
25330 1999 KV4      2.70 0.080 0.031 1.30 1.495 16 4.919 02-06-2003 - d 
25330 1999 KV4      3.21 0.050 0.020 1.50 1.197 54 4.919 10-05-2001 - e 
33342 1998 WT24     0.34 0.590 0.232 0.90 0.990 67 3.697 18-12-2001 - d 
33342 1998 WT24     0.44 0.350 0.137 1.50 0.987 79 3.697 19-12-2001 - d 
33342 1998 WT24     0.50 0.270 0.106 1.85 0.981 93 3.697 21-12-2001 - d 
35396 1997 XF11     0.89 0.320 0.126 1.30 1.215 30 3.257 28-11-2002 - d 
35396 1997 XF11     0.91 0.310 0.122 1.20 1.018 53 3.257 03-11-2002 - d 
35396 1997 XF11     1.18 0.180 0.071 1.80 1.034 63 3.257 05-11-2002 - d 
53789  2000 ED104 1.20 0.180 0.071 1.68 1.089 60      - 29-09-2002 - g 
85953  1999 FK21    0.59 0.320 0.126 0.91 1.140 35      - 21-02-2002 - e 
86039  1999 NC43    2.22 0.140 0.055 2.86 1.116 59 34.49 17-03-2000 - e 
99935  2002 AV4     1.50 0.370 0.145 1.60 1.086 70      - 01-06-2003 - d 
  1999 HF1     4.74 0.110 0.043 1.68 0.957 91      - 22-03-2002 - g 
  2000 BG19    1.77 0.040 0.016 0.74 1.388 17      - 17-03-2000 - e 
  2001 LF       2.00 0.050 0.020 1.40 1.172 45      - 02-06-2003 - d 
  2002 BM26    0.84 0.020 0.008 3.10 1.023 60      - 21-02-2002 - e 
  2002 HK12    0.80 0.170 0.067 2.84 1.138 33      - 28-09-2002 - g 
  2002 NX18    2.40 0.030 0.012 1.19 1.158 54      - 29-09-2002 - g 
  2002 QE15    1.94 0.150 0.059 1.53 1.131 62      - 28-09-2002 - g 
   2003 YT1     1.50 0.270 0.106 1.92 1.035 74      - 08-05-2004 - d 
Table 1 Near Earth-asteroids with η-values derived from spectral fitting to multi-wavelength mid-
infrared observations. The object effective diameter, D, the geometric visible albedo pV, and the η-values 
have been derived by using the NEATM. α is the phase angle at the epoch of the observations, which is 
given in the “Obs. Date” column. P is the rotation period in hours. In the column “Spin axis” a “Y” 
indicates that the spin axis orientation of the asteroid is known. In the column “Re” we give the original 
publication reference: a) Harris and Davies (1999); b) Harris et al. (2005); c) Harris (1998); d) Delbo’ 
(2004); e) Delbo’ et al. (2003); f) Harris et al. (1998); g) Wolters et al. (2005). 
- 25 - 
Figure captions 
Fig. 1 Sensitivity of η to model parameter variations. Δη/η (%) caused by a change of 
1% in the bolometric Bond albedo A (upper left), macroscopic surface roughness 0θ  
(upper right), heliocentric distance r (lower left), and the thermal parameter Θ (lower 
right). See text, section 2.1 for details. 
Fig. 2 Distribution of the input parameters used in our statistical inversion method. 
Upper left: distributions of NEA rotation rates from “Physical parameters of NEOs 
(Hahn, 2006, http://earn.dlr.de”). Upper right: distribution of NEA phase angles. Lower 
left: distribution of NEA sub-solar latitudes θSS. Lower right, solid line: distribution of the 
geometric albedos (pV) and, dashed line: the bolometric Bond albedos (A) for the 
asteroids of Table 1 having η-values determined from observations. 
Fig. 3 Dependence of η-value on phase angle, α. Black diamonds: η-values derived 
from the NEATM for a set of NEAs with adequate multi-filter photometric data to enable 
η to be derived via spectral fitting (the data set includes multiple values of η for some 
objects observed at more than one phase angle; for the original data sources see Table 1). 
The error bars represent a 20% uncertainty, which is based on the reproducibility of η for 
those objects for which more than one measurement is available from independent data 
sets. Colored points: distributions of (α, η) calculated by means of our model for different 
values of thermal inertia: i.e. 15 (green), 200 (red), and 1000 (blue) J m-2 s-0.5 K-1. The 
distribution of the measured η-values is best described by the red points. 
Fig. 4 Plot of the function D (Γ), i.e. the distance D of the two-dimensional 
Kolmogorov-Smirnov best-fit procedure against the thermal inertia Γ. The three curves 
were generated assuming three different values of the surface roughness: solid line 
θ =58°; dotted line θ =36°; dashed line θ =0° i.e. a smooth surface. The dashed-dotted 
line shows the function D (Γ) obtained by using θ =58° and a random distribution of 
asteroid spin-axis orientations uniformly distributed over the celestial sphere, instead of 
the nominal one, as input for our model. 
Fig. 5 Histogram of the distribution of the log of the diameters, D, of the NEAs for 
which we have η-values determined from observations. The best-fit Gaussian function, 
0.37exp(-z2/2), where z=(log D - 0.23)/0.31, with D in km, is also shown. 
Fig. 6 Thermal inertia as a function of asteroid diameter. Small open circles represent 
values from the literature derived by means of thermophysical models. The large open 
diamond is the result from this work (see text for details). The straight (continuous) line 
which gives the best fit to the trend of increasing thermal inertia, Γ, with decreasing 
asteroid diameter, D, is given by the expression Γ=300×D-0.48. The axis on the right-hand 
side gives the asteroid surface thermal conductivity k on the basis of k=Γ2/(ρc), assuming 
constant surface density, ρ, equal to 2500 kg m-3 and specific heat capacity, c, equal to 
600 J kg-1 K-1. These values are reasonable assumptions for asteroid surfaces (Britt et al., 
2002; Farinella et al., 1998). The thermal conductivities of (6489) Golevka (Chesley et 
al., 2003) and for Karin cluster asteroids (Nesvorný and Bottke, 2004) are indicated with 
arrows. The two values of Γ derived for 2002 NY40 are indicated as the lower and the 
upper limits of the error bar on the extreme left of the plot. Dotted line: linear regression 
of Eq. (6) for MBAs only; dashed line: linear regression of Eq. (6) for NEAs only. 
- 26 - 
Figures 
0 20 40 60 80 100
α (deg)
0 20 40 60 80 100
α (deg)
0 20 40 60 80 100
α (deg)
0 20 40 60 80 100
α (deg)
Fig. 1 
- 27 - 
0 1 2 3 4 5 6 7 8 9 10 11 12
NEA rotation rate, P (hours)
0 10 20 30 40 50 60 70 80 90 100
Solar phase angle, α (degrees)
0 10 20 30 40 50 60 70 80 90 100
Sub-solar latitude, θSS (degrees)
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Albedo
Fig. 2 
- 28 - 
Fig. 3  
- 29 - 
0 100 200 300 400 500 600 700 800 900 1000
Thermal Inertia (J m-2 s-0.5 K-1)
Fig. 4  
- 30 - 
-2 -1 0 1 2 3
log D (D in km)
Fig. 5  
- 31 - 
1 10 100 1000
Asteroid diameter (km)
1 10 100 1000
Asteroid diameter (km)
Fig. 6
