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Appendix A: Thermal modelling of WISE asteroid data

Our aim is to model the observed asteroid flux as a function of several physical parameters and derive the set of parameter values that most closely reproduce the actually measured fluxes. In this work we follow the method described by Mainzer et al. (2011b). The set of wavelengths covered by WISE (specified in Sect. 2) allow us to derive up to three parameters by fitting a thermal model to asteroid WISE data: asteroid effective diameter, beaming parameter, and reflectance at 3.4 μm (defined below). Within the wavelength range covered, the observed asteroid flux consists of two components: (A.1)The thermal flux component (f_th,λ) is the main contribution to W3 and W4, whereas the reflected sunlight component (r_s,λ) dominates in band W1. In general, W2 will have non-negligible contributions from both components (Mainzer et al. 2011b).

The computation of f_th,λ is based on the Near Earth Asteroid Thermal Model (NEATM; see Harris 1998; Delbó & Harris 2002). The asteroid is assumed to be spherical, and its surface is divided into triangular facets that contribute to the total thermal flux observed by WISE in accordance with the facet temperature (T_i), the geocentric distance (Δ), and the phase angle (α_⊙). In turn, the temperature of each facet depends on the asteroid heliocentric distance (r_⊙) and its orientation with respect to the direction towards the sun. It is given by (A.2)which results from assuming that each surface element δa_i is in instantaneous equilibrium with solar radiation. S_⊙ is the solar power at a distance of 1 AU, A the bolometric Bond albedo, ϵ the emissivity (usually taken to be 0.9; see Delbó et al. 2007, and references therein), σ the Stefan-Boltzmann constant, and μ_i = cosθ_i, where θ_i is the angle between the normal to the surface element i and the direction towards the Sun. Non-illuminated facets will be instantaneously in equilibrium with the very low temperatures of the surroundings (~0 K), and thus their contribution to f_th,λ is neglected in the NEATM. Finally, the beaming parameter (η) can be thought of as a normalisation or calibration factor that accounts for the different effects that would change the apparent day-side temperature distribution of the asteroid compared to that of a perfectly smooth, non-rotating sphere (Harris 1998). These include, for example, the enhanced sunward thermal emission due to surface roughness (η < 1), or the non-negligible night-side emission of surfaces with high thermal inertia that, in order to conserve energy, causes the day-side temperature to be lower than that compared to the ideal case with zero thermal inertia (η > 1).

The asteroid thermal flux component is then given by (A.3)where f_i,λ is the contribution from each illuminated facet of a 1-km sphere; Ω ≡ (D/1   km)² scales the cross-section of the latter to the corresponding value of an asteroid of diameter D. The colour correction associated with each value of T_i and each WISE band is applied to the facet flux. By definition, it is the quotient of the in-band flux of the black body at the given temperature to that of Vega (Wright et al. 2010). A colour correction table was generated for all integer temperatures from 70 K up to 1000 K using the filter profiles available from Cutri et al. (2012).

The reflected light component, the second term on the right-hand side of Eq. (A.1), is calculated as follows. First, the asteroid visible magnitude (V) that would be observed at a given geometry (r_⊙, Δ and α_⊙) can be estimated using the IAU phase curve correction (Bowell et al. 1989), along with the tabulated values of asteroid absolute magnitude (H) and slope parameter (G) from the Minor Planet Center. Secondly, knowledge of the solar visible magnitude and flux at 0.55 μm (V_⊙ and f_V⊙, respectively) allows us to calculate the sunlight reflected from the asteroid at that particular wavelength: (A.4)If we assume that the Sun is approximated well by a black-body emitter at the solar effective temperature (T_⊙ = 5778 K), the estimated reflected flux at any other desired wavelength (r_λ) can be computed by normalising the black body emission B_λ(T_⊙) to verify r_V, i.e. (A.5)In this approximation, we can also consider (A.6)from which we arrive at the following expression: (A.7)where the subscript IR denotes 3.4 μm. We do not colour-correct this component given the small correction to the flux of a G2V star (see Table 1 of Wright et al. 2010). Finally, to account for possible differences in the reflectivity at wavelengths longward of 0.55 μm, a prefactor to r_λ is included in the model, such that (A.8)This prefactor, R_p, is by definition equivalent to the ratio of p_IR and the the visible geometric albedo, so we will refer to it as the “albedo ratio”. The paremeter p_IR is the reflectivity at 3.4 and 4.6 μm defined by Mainzer et al. (2011b).

To sum up, the observed model flux can then be written as (A.9)We use the Levenberg-Marquardt algorithm (Press et al. 1986) in order to find the values of asteroid size (, in km), beaming parameter (η) and albedo ratio (R_p) that minimise the χ² of the asteroid’s WISE data set, namely (A.10)where F_j,λ and σ_j,λ are the measured fluxes and corresponding uncertainties, j runs over the observation epochs, and λ labels the WISE bands. The implementation of this technique involves calculating the partial derivatives of with respect to the fitting parameters, which is straightforward in the case of Ω and R_p. The partial derivative with respect to η can be derived from (A.11)

Appendix B: Comparison with Masiero et al. (2011)

	Fig. B.1 Fractional difference histograms of D, η, pV, and Rp. We define ε = 100(x − xM)/x, where x is the parameter value in this work and xM the correspoding value taken from Table 1 by Masiero et al. (2011). The vertical lines mark the corresponding average values. Only parameters resulting from the same input values of H contribute to these histograms.
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Figure 2 shows that our parameter determinations and those of Masiero et al. (2011) are compatible in spite of the slight differences in the data set and the thermal modelling used in this work (refer to Sect. 2 and Appendix A), from which we do not expect to obtain exactly the same best-fit parameters for each object. In order to carry out a detailed comparison between our results and those of Masiero et al. (2011), we computed the mean fractional difference (ε) and corresponding standard deviations of D, η, p_V, and R_p. Let ε = 100(x − x_M)/x, where x is the parameter value for a given object in this work, and x_M the correspoding value taken from Table 1 by Masiero et al. (2011). The distributions of ε values are plotted in Fig. B.1. These histograms only include parameter determinations that have the same H as input in order to identify possible discrepancies in results not caused by different values of H. We find that our values of D and η tend to be slightly higher by 1% and 3%, respectively, whereas our p_V values are lower by 2%, though these deviations are small compared to the error bars. On the other hand, there is a large bias towards lower values of R_p that, while still being within the error bar, must be addressed.

Most probably, the R_p discrepancy is associated with how the reflected flux r_λ is calculated. In particular, we take the solar flux at 3.4 μm (f_IR⊙ in Eq. (A.7)) from the solar power spectrum at zero air mass of Wehrli⁴, based on the one by Neckel & Labs (1984). Any differences in input, including solar visible magnitude, taken from tabulated data sources that may cause our r_λ to be systematically 10% greater than that of Masiero et al. (2011) would explain our higher values of R_p. For instance, considering that there is only one optimum value of r_s,λ to fit a given W1 data set, from Eq. (A.8) it is clear that larger r_λ will have associated a lower best-fit value of R_p.

	Fig. B.2 Differences in albedo ratio determinations versus difference in absolute magnitude corresponding to the B-types in this paper and those by Masiero et al. (2011).
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The Monte Carlo estimations by the NEOWISE team show that the error bars associated to the fitting of the data are always small compared to the errors inherent to the thermal model itself. The relative errors in diameters derived from the NEATM have been characterised as  ~10%–15% (Harris 2006). From these facts and the widths of the ε-value distributions of Fig. B.1, we consider it safe to assume a minimum relative error of 10% in diameter and 20% in beaming parameter, p_V and R_p. On the other hand, large uncertainties in the absolute magnitude (sometimes as large as  ~0.3 mag) will also affect the values of p_V, so 20% is probably an optimistic assumption in some cases.

Finally, we also evaluate how differences in the values of H result in different values of p_V and R_p. We downloaded the MPC orbital element file as of May 2012 and compared the values of absolute magnitude (H_U) to those used by Masiero et al. (2011), H_M. About 50000 H-values have been updated between these two works, and  ~38 000 have been enlarged. Figure 1 shows a histogram of ΔH ≡ H_U − H_M for the B-types in this work. Out of the 52 objects with ΔH ≠ 0, as many as 43 of them have ΔH > 0. Our size determinations agree to within 10%, therefore higher updated values of H will result in lower values of geometric albedos.

In Fig. B.2 we show a plot of ΔR_p ≡ R_p − (R_p)_M versus ΔH for all the B-types with determined values of R_p. The notation (R_p)_M refers to the corresponding albedo ratios by Masiero et al. (2011). There are three features to note in this plot: (1) our values of R_p tend to be  ~10% systematically lower, as we already noted (see Fig. B.1); (2) most points off the ΔH = 0

axis show a direct correlation between ΔR_p and ΔH, as expected from the discussion above; (3) some points show ΔR_p <  −0.5, even though ΔH = 0. The points of feature (3) are explained by an inconsistency in the p_V values of Masiero et al. (2011) with their corresponding values of D and H: they do not verify Eq. (1) and are always lower than the predicted p_V.

To sum up, we have shown that if the input values of H are equal, our model fits are consistent within the model error bars with those presented in Table 1 of Masiero et al. (2011). The tendency to 10% lower values of R_p is likely caused by differences in solar power spectra data taken to estimate the reflected light component at NIR wavelengths (see Eq. (A.7)). We have also examined how updated input values of H affect the best-fit parameter values and showed how increasing the value of H results in greater values of R_p and vice versa.

© ESO, 2013