Thermal inertia of near-Earth asteroids and implications for the magnitude of the Yarkovsky effect
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 , 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): The subsolar temperature, , is determined by equating the total energy absorbed by a surface element to that emitted in the thermal infrared, i.e. where A is the bolometric Bond albedo, 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 obtains for on a spherical surface 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 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, , and consequently depends on the heliocentric distance of the body. The thermal parameter is given by 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 also offers an explanation for 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, Christensen, 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 12,000 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 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, 2006). 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, , according to the parameterization introduced by Hapke (1984), emission spectra are a function of only and not of the crater opening angle and crater surface density. We recall here that where θ is the angle of a given facet from the horizontal, and 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, ) are adjusted until the best agreement is obtained with the observational data, i.e., the least-squares residual of the fit, , 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 and Lagerros, 1998, Müller and Blommaert, 2004, Mueller et al., 2006), and five NEAs (Harris et al., 2005, Harris et al., 2007; Müller et al., 2005, Mueller 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 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.
Section snippets
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
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 et al., 2007), (1580) Betulia (Harris et al., 2005), (25143) Itokawa (Mueller et al., 2007, Müller 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 et al. (2005) have obtained a thermal
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, , is proportional to (Bottke et al., 2002). However, the mean value of the thermal inertia derived for NEAs and the
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
Acknowledgements
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.
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