© ESO, 2013

1. Introduction

The study of asteroids is important for gaining knowledge about the origin and evolution of our planetary system. Asteroids are relics of the solar system’s formation and the building blocks of the terrestrial planets. Primitive asteroids, i.e. those belonging to the so-called spectroscopic C-complex and having in general visible geometric albedo p_V ≲ 0.1 and featureless, flat visible spectra, are particularly relevant in this context. Thought to have formed farther away from the Sun than the other asteroid classes, primitive asteroids have experienced less heating and alteration processes and have a more pristine composition, potentially preserving crucial information about the early solar system. In addition, primitive asteroids play an important role in current exobiological scenarios since they delivered complex organic molecules to the early Earth. This organic matter is a prerequisite for the synthesis of pre-biotic biochemical compounds that would subsequently lead to the emergence of life (Maurette 2006, and references therein). For these and other reasons, upcoming sample return space missions have selected primitive asteroids as primary targets: NASA’s OSIRIS-Rex (Campins et al. 2010b; Lauretta et al. 2010), ESA’s Marco Polo-R (Barucci et al. 2012; de León et al. 2011), and JAXA’s Hayabusa-2¹.

The taxonomic classification of primitive asteroids has been traditionally based on their low visible albedo ( ≲ 0.08–0.1), relatively flat or slightly blue visible spectra with weak or no absorption features (for a detailed review, see Clark et al. 2010). Several primitive classes have been defined in Tholen’s taxonomy, e.g. B, C, F, G, D, and P (Tholen 1984, 1989). Bus’s feature-based classification, independent of the albedo, merged some of these and defined new primitive taxons that were extended with minor changes into the Bus-DeMeo taxonomy (Bus & Binzel 2002; DeMeo et al. 2009).

B-types are of particular interest among the primitive asteroids for a number of reasons: (i) there is as yet no compelling explanation for their defining feature, i.e. their slightly blue spectral slope in the visible range; (ii) B-type asteroids constitute the only primitive class that presents a wide range of spectral slopes in the 0.8–2.5 μm near-infrared (NIR) region (de León et al. 2012); (iii) the few B-types studied present the 3-μm absorption feature related to hydrated minerals; (iv) water ice has been detected on the surface of (24) Themis (Campins et al. 2010a; Rivkin & Emery 2010); (v) the majority of asteroids that have been observed to display cometary-like activity are B-types (Licandro et al. 2012, and references therein); (vi) the target of NASA OSIRIS-Rex mission, 2006 RQ₃₆, is a B-type asteroid.

The B-type asteroids have been widely related to carbonaceous chondrites, composed of carbonaceous minerals and phyllosilicates, in terms of their generally low albedo and broad spectral properties (Gaffey et al. 1989; Vilas & Gaffey 1989; Vilas et al. 1994). More recently, de León et al. (2012) have examined visible to near-infrared (VNIR) spectra of a sample of 45 B-types and found that the characteristic negative spectral slope in visible wavelengths diverges into a continuum of gradually varying NIR spectral slopes, from a monotonic negative (blue) slope to a positive (red) slope. De León et al. classified their spectra into six “average spectra” or “centroids” representative of the whole sample by means of statistical clustering analysis (Marzo et al. 2009). These centroids were compared against meteorite spectra from the RELAB database (Pieters & Hiroi 2004). The best meteorite analogues found for the six “clusters” were all carbonaceous chondrites with a gradual change in their degree of hydration, from aqueously altered CM2 chondrites for the reddest cluster to the heated/thermally metamorphosed CK4 chondrites for the bluest one.

This work is an extension of the results obtained by de León et al. (2012), which are part of a ongoing programme devoted to improving our knowledge of B-types. Our aim here is to study the physical properties of B-type asteroids that can be derived by fitting a thermal model to their NASA’s Wide-field Infrared Explorer (WISE) observations, i.e. effective diameter, beaming parameter, and p_IR/p_V, where p_IR is the albedo at 3.4–4.6 μm as defined in Mainzer et al. (2011b). We closely follow the methodology of Mainzer et al. (2011b), though with a number of differences, as described in Sects. 2 and 3.

Mainzer et al. (2011c) studied all groups of spectrophotometrically classified asteroids in the Tholen, Bus, and Bus-DeMeo taxonomies observed by WISE, including B-types. Those authors present visible and NIR geometric albedo distributions and median values of B-types and conclude that, in spite of having analogously low albedos, B-, C-, D-, and T-type asteroids can be distinguished from their values of NIR reflectance. In particular, Mainzer et al. (2011c) point out that B-types have a lower p_IR/p_V ratio than C-types and attribute this to their blue VNIR slopes likely extending out to 3–4 μm. Our definition of B-type asteroid in this work is different: following Clark et al. (2010) and de León et al. (2012), we consider all objects that have a flat to slightly blue spectral slope in the visible range, i.e. any object that has ever been classified as B-type, including Tholen’s F-types and ambiguous designations. This criterion produces a total of 162 asteroids classified as B-types.

We also study the collisional family of (2) Pallas (hereafter PCF and Pallas, respectively). The PCF is interesting for the following reasons: (i) it is a B-type family, given that Pallas and the very few family members that have been taxonomically classified are B-types (Gil-Hutton 2006; de León et al. 2010), and the five members studied in de León et al. (2012) were spectrally related to carbonaceous chondrites, which establishes their primitive nature; (ii) the average values of geometric albedo of members of the family calculated by Masiero et al. (2011) are roughly 0.15 (see their Fig. 19), significantly greater than expected for primitive bodies (<0.1), though no explicit comment is made by these authors on this intriguing result; (iii) the near-Earth asteroid (3200) Phaethon, an activated asteroid parent of the Geminid meteor shower, likely originated in the Pallas family (de León et al. 2010); (iv) this family is isolated well in (proper) element space, thus the potential identification of interlopers as members is greatly reduced.

The paper is organised as follows. In Sect. 2, we briefly describe the WISE data set and our selection criteria. The thermal modelling of the data is explained in detail in Appendix A, whereas Sect. 3 includes relevant comments on the very few differences introduced in this work. In Appendix B our parameter determinations are compared to those by Masiero et al. (2011). We present our results in Sect. 4, a discussion of the implications of this work is put forward in Sect. 5, and our conclusions are enumerated in Sect. 6.

2. Data

A general introduction to WISE can be found in Wright et al. (2010) and references therein. Of particular interest to solar system science is the NEOWISE project. This acronym collectively refers to two enhancements to the WISE data processing system that were designed to allow detection and archiving of solar system objects (for details, see Mainzer et al. 2011a).

WISE used four broad-band filters with approximate isophotal wavelengths at 3.4, 4.6, 12, and 22 μm, referred to as W1, W2, W3, and W4, respectively (Wright et al. 2010). The WISE All-Sky Single Exposure L1b Working Database, published in April 2012 and available via the IRSA/IPAC archive², includes the corresponding magnitudes and uncertainties in the Vega system, as well as quality and contamination and confusion flags that enable us to reject defective data (Cutri et al. 2012).

We follow a combination of criteria found in Mainzer et al. (2011b,c), Masiero et al. (2011), and Grav et al. (2012) to ensure the reliability of the data. We implement the correction to the red and blue calibrator discrepancy in W3 and W4, and we use a cone search radius of 0.3′′  centred on the MPC ephemeris of the object in our queries. All artefact flags other than p, P, and 0 and quality flags other than A, B, and C are rejected, and we require the modified Julian date to be within four seconds of the time specified by the MPC and split groups of epochs separated more than three days (see the end of this section). We ensure that the data is not contaminated by inertial sources by removing those points that return a positive match from the WISE Source Catalog within 6′′. Finally, all remaining observations in a given band are rejected if they are fewer than 40% of the data in the band with the maximum number of detections.

On the other hand, we do not use data saturated to any extent. The onset of saturation is reported to correspond to magnitudes M_W1 < 6, M_W2 < 6, M_W3 < 4, M_W4 < 3 (Cutri et al. 2012). We found that enlarging the error bar of partially saturated data to 0.2 magnitudes (which translates into a relative error of 20% in fluxes) renders the corresponding band unable to play any effective role in the thermal model fit by not contributing significantly to the χ².

The application of the above criteria results in a sample of 111 B-type main-belt asteroids with WISE observations usable for our purposes. Some of these have been observed by WISE in more than one uninterrupted group of epochs with different observation geometries. We also model such groups of observations separately if they are more than three days apart (see Appendix A and Mainzer et al. 2011b). Consequently, we have a larger set of parameter determinations than asteroids in the sample.

3. Thermal modelling

The modelling of WISE asteroid data implemented in this work closely follows Mainzer et al. (2011b,c), and Masiero et al. (2011) and is based on the near-Earth asteroid thermal model (NEATM, Harris 1998) and the IAU phase curve correction to the visible magnitude (Bowell et al. 1989). For the sake of reproducibility, we include a detailed account of our procedure in Appendix A and enumerate the few differences with respect to Masiero et al. and Mainzer et al. below.

The number of parameters we can fit for each object depends upon how many and which WISE bands are present in its data set. Parameter default values are chosen based on the peak of their respective fitted value distributions of main belt asteroids presented in Masiero et al. (2011). Whenever there is one single or no thermal band available (W2, W3, or W4) we assume η = 1.0; R_p is fixed to 1.5 unless we have at least a 50% contribution of reflected sunlight in W1 data. The last criterion is based on the consistency of our parameter determinations for objects with double detections³; namely, both groups of observations of asteroids (1076) and (2446) have >50% sunlight and their R_p values are consistent within the error bar. On the other hand, (3579) has incompatible R_p determinations from W1 data with  >70% and  ~25% reflected sunlight, respectively. We thus reject three R_p values belonging to asteroids (288), (1493), and (3579).

These considerations allow asteroid size to be fitted in all cases and, by means of the relation ${\mathit{p}}_{\mathrm{V}}\mathrm{=}{\left(\mathrm{1329}\hspace{0.17em}\left[\mathrm{km}\right]\frac{{\mathrm{10}}^{\mathrm{-}\mathit{H}\mathit{/}\mathrm{5}}}{\mathit{D}\hspace{0.17em}\left[\mathrm{km}\right]}\right)}^{\mathrm{2}}\mathit{,}$(1)the geometric visible albedo can be computed. In contrast, W1 and W2 data are more often rejected based on the data requirements (see Sect. 2) than the purely thermal bands, and one will usually be able to obtain fewer R_p determinations than η or indeed D.

It is important to point out that we do not use physical data previously determined by direct measurements, such as radar diameters or albedos, to constrain our fits; i.e., we limit ourselves to using radiometrically derived sizes. This will introduce variations in the parameter determinations of some individual asteroids – especially the largest ones, since it is more likely that more direct measurements have been performed – as compared to Masiero et al. (2011), but should not affect the result of statistical analyses if the populations studied consist of a significant number of objects. In Appendix B we show that, given the same input values of H, our best-fit parameter values are consistent within the error bars, though we find that our R_p determinations are systematically lower by  ~10%. On the other hand, the update of  ~45% of the MPC H-values in our sample does change the values of p_V (see Eq. (1)) and R_p. As shown in Fig. 1, the updates tend to be toward higher values of H, which will result in lower p_V and higher values of R_p than those of Masiero et al. (2011). For more details, see Appendix B.

Fig. 1 Differences in absolute magnitude values between those used by Masiero et al. (2011), HM, and the most updated ones (as of May 2012) used in this paper, HU. Note that the cases verifying ΔH = 0 do not contribute to this histogram.

4. Results

4.1. Value distributions of η, p_V, and R_p

The distributions of η, p_V, R_p, and p_IR obtained for the B-types are shown in Fig. 2 (complete set of parameter determinations in Table 1). For comparison, we overplotted the corresponding histograms with best-fit parameter values from Masiero et al. (2011) (see Appendix B for a detailed comparison).

Fig. 2 Histograms of value distributions obtained for the B-type asteroids observed by WISE: η (top left), pV (bottom left), Rp (top right), and pIR (bottom right). The corresponding histograms using the results from Masiero et al. (2011) are overplotted.

The parameter median and mean values and standard deviations of this work, as well as the number of parameter determinations obtained in each case (N), are presented in Table 2. Note that p_V and p_IR are not fitted, but computed. The former is obtained from Eq. (1) with the best-fit value of D as input, whereas p_IR = R_pp_V. These results are consistent with previous work by Mainzer et al. (2011c): if we take the weighted mean of median p_V- and R_p-values corresponding to their Tholen, Bus, and Bus-DeMeo B-types and Tholen F-types, we obtain the same median values.

We find a η-value distribution centered at unity, consistent with the average value obtained for the whole main belt (Masiero et al. 2011). The broad and asymmetrical p_V distribution extends to p_V > 0.1. The R_p distribution is also broad, whereas the values of p_IR are more compactly distributed around the mean.

4.2. Albedo ratio

Figure 3 shows a plot of R_p versus p_V. Similar plots including all taxonomic classes in different classification schemes are presented in Figs. 14 and 15 by Mainzer et al. (2011c) to show how clearly different taxons may be distinguished. Here we concentrate on the p_V < 0.18 range, with all B-types with WISE data for which R_p-values could be derived are plotted (see Table 2); we also include all main belt asteroids taken from Table 1 of Masiero et al. (2011). The cloud of points exhibits a characteristic “waning-moon” shape, with no points in either the upper right- or lower left-hand regions of the plot. Mainzer et al. caution that, while WISE is essentially unbiased against p_V, spectroscopic surveys conducted to create the classification schemes are inherently biased against small, low-p_V objects; in addition, the computation of R_p from WISE data requires sufficient reflected sunlight contribution in bands W1 and W2, which will tend to exclude objects with low enough values of p_V and R_p. This could explain the lack of points in the lower left-hand part of the plot. However, these biases cannot be solely responsible for this characteristic shape since other taxonomic classes with higher values of p_V and R_p also cluster similarly. Furthermore, if we plot all main belt objects irrespective of whether they have a taxonomic classification or not, distinct clouds of points with the same shape become apparent. Thus, we emphasise that because of their characteristic p_V- and p_IR-value distributions, there are no high-R_p objects among the high-p_V B-type asteroids.

Fig. 3 Albedo ratio versus visible geometric albedo. The 20% error bars in pV and Rp are not shown to facilitate visualisation. B-types observed by WISE are plotted in black circles; all main belt objects featured in Table 1 of Masiero et al. (2011) are in empty grey circles. The horizontal line is an artificial feature corresponding to objects with the default fixed value of Rp = 1.5.

We have also analysed the R_p-values of the sample of 45 B-types studied by de León et al. (2012) separately. The spectra of these asteroids were classified into six “average spectra” or “centroids” referred to as G1, G2, ... G6 (see Sect. 1). These show a progressive decrease in spectral gradient in the NIR interval (0.8–2.5 μm), ranging from a positive (red) slope for G1 to a negative (blue) slope for G6. In Fig. 4, we plot R_p versus p_V labelling the objects in the different centroids G1, ... G5. Cluster G6 is not included since it is only composed of one member, (3200) Phaethon, which did not have enough WISE observations to perform a reliable fit.

Fig. 4 Rp vs. pV for the asteroids in de León et al. (2012) distinguishing the clusters to which they belong. Within parentheses, the average pV of each cluster is shown. On average, pV increases from G1 to G3, keeping the same value from G3 to G5.

The average values of p_V for each cluster increases from G1 to G3, retaining the same value from G3 to G5. This might suggest an inverse correlation between the cluster NIR slope and p_V, though the small number of objects per cluster with WISE observations (2 objects in G1, 12 in G2, 6 in G3, 5 in G4, and 3 in G5) prevents establishing a firm conclusion.

In Fig. 5, the average R_p-value is plotted for the different clusters.

Fig. 5 Average values of Rp for each cluster defined by de León et al. (2012). In terms of spectral slope up to 2.5 μm, cluster G5 is the bluest, whereas G1 is the reddest. The error bar is the standard error of the mean.

This figure suggests a correlation between the average R_p and the NIR slope of the relative reflectance of the clusters: objects with higher R_p belong (on average) to clusters with higher NIR spectral slope in the  ~ 1.0–2.5 μm range (see also Fig. 5 of de León et al. 2012). This indicates that the reflectivity at 3.4 μm tends to continue the trend observed at shorter IR wavelengths, as hypothesised by Mainzer et al. (2011c).

The W1 band pass spans the 2.8 to 3.8 μm range (Wright et al. 2010); therefore, the values of R_p may also indicate the presence of the 3-μm absorption feature attributed to hydrated minerals or water ice detected on many asteroids (Rivkin et al. 2000; Gaffey et al. 2002; Campins et al. 2010b; Rivkin & Emery 2010; Licandro et al. 2011). Mainzer et al. (2011c) ruled out the possibility of detecting the hydration band from WISE data based on the fact that the average R_p for a sample of 7 M-types with positive detections of the band (Rivkin et al. 2000) cannot be distinguished from those corresponding to other 33 M-types. However, this test might not be meaningful given that the R_p-values have error bars that are at least larger than the characteristic depth of the absorption feature. Below we provide evidence that the W1 may be sensitive to the 3-μm feature.

Our first step was to assemble VNIR spectra up to  ~ 3.6   μm of a list of nine C-complex control asteroids. All of these asteroids are primitive, including some B-types, such as (2) Pallas or (45) Eugenia, and some of them show a distinct absorption feature. The assembled spectra are plotted along with the best-fit value of R_p in Fig. 6. We collected or digitised data from Hiroi et al. (1996), SMASS-II (Bus & Binzel 2002), the 52-Color Survey (Bell et al. 2005), Rivkin et al. (2003) and de León et al. (2012).

Fig. 6 Assembled spectra of the control objects chosen to study the reflectivity at 3.4 μm derived from WISE data as a diagnostic of the presence of the 3-μm feature. The dashed lines approximately enclose the wavelength integration range of the W1 filter.

Taking into account the errors associated with superimposing spectra obtained at different epochs of observation with different equipment and the uncertainties in the spectra and in the value of the albedo ratio, the latter is not expected to exactly match the value of the spectra at 3.4 μm. Owing to these deviations, seen in Fig. 6, one can neither confirm nor rule out the presence of the absorption feature based on the value of R_p alone.

Next, we combined the values of R_p with the observed relative reflectances at 2.5 μm available from other datasets. Rivkin et al. (2003) use the parameter 1 − R_λ/R_2.5 as a rough measure of band depth. Because the band minima are usually near 3.0 μm and given that W1 results from an average over 2.8 to 3.8 μm, the parameter b ≡ 1 − R_p/R_2.5 is not to be taken as a measure of band depth but as a helpful parameter to quantitatively compare the values of relative reflectances at 2.5 and 3.4 μm. If a deep absorption band is present, R_p is in general expected to be  ≲ R_2.5, hence b ≳ 0, as is the case for (19) Fortuna. There is one general case for which this interpretation would be wrong: if the NIR slope is negative up to 3.4 μm, we would have b ≳ 0 even if no band was present, though blue NIR slopes such as Pallas’ have only been measured in a low percentage of cases (e.g. 2 out of 45 in the sample of de León et al. 2012).

A plot of b vs. p_V is shown in Fig. 7 with an estimated b error bar of 0.2 to account for the large uncertainties in R_p and the assembly of the spectra. We see that those asteroids with a weak or non-existent absorption band tend to have b ≲ 0, whereas b ≳ 0 for those with a higher contrast feature. Only in the case of (13) Egeria is the value of b clearly inconsistent with the absortion band observed. Indeed, the large error bar in the value of b still prevents irrefutable detection of the band on a case-by-case basis, but from Fig. 7, the correlation between the sign of b and the spectra appears robust, so that systematically obtaining b ≳ 0 for a given population may be statistically significant.

In Fig. 8 (top panel) we show a b-value histogram of those B-type asteroids for which the value of b could be determined, and R_2.5 was taken from the B-type spectra presented in de León et al. (2012), after normalising to unity at 0.55 μm. To test for statistical significance, we resort to the Kolmogorov-Smirnov (KS) test, which enables one to reject the null hypothesis that a given set of unbinned values is compatible with having been drawn from a given distribution function (see e.g. Press et al. 1986). As the null hypothesis we take a Gaussian distribution with zero mean. This choice is based on the mean value of b that we would expect considering that interpolating between R_0.55 = 1 and gives (see Fig. 9). The KS test amply rules out the null hypothesis that the B-types b-values are drawn from a Gaussian distribution of zero mean regardless of its width (σ).

To demonstrate the robustnest of this result even further, we carried out the same procedure for a list of asteroids belonging to the S-complex, including all Bus-DeMeo pure S-types for which we found that the value of R_2.5 was available from the literature and for which R_p could be computed, plus enough randomly selected S subtypes to get the same number of b-values as we derived for the B-types (see Table 5). S-type asteroids are “anhydrous” and have positive spectral slopes up to 2.5 μm so that in the absence of a 3-μm absorption feature, one would expect a negative value of b if the spectral slope maintains its trend up to 3.4 μm. Taking into account that for this sample of S-types, from the interpolated value (see Table 3), the expected mean value of b would be (see Fig. 9). The distribution of b-values obtained for the S-types, shown in the bottom panel of Fig. 8, presents a negative mean value. In this case, the KS test does not rule out the null hypothesis that the b-values are drawn from a Gaussian distribution centered at  − 0.14 with σ = 0.2 with p-value >0.9. This rules out the possibility that a systematic error is causing the b-values of the B-type asteroids to be >0.

In the preceeding section we have shown that our values of R_p are systematically  ~ 10% lower than those of Masiero et al. (2011). We carried out the same procedure enlarging our R_p values 10% and the conclusions still hold.

To conclude, we find that the majority of B-types with computed b-values verify b > 0 and that very few present a clearly negative value of b, which means that even for those B-type spectra with a positive slope in the 2.5 μm region (approximately half of the objects in de León et al. 2012), there is a reduction in the reflectivity around 3.4 μm. As discussed above, for asteroids of the C-complex, the 3-μm absorption feature has been attributed to hydrated minerals or water ice. The presence of goethite has been also proposed as an alternative explanation for this band (Beck et al. 2011). Nonetheless, while other closely related minerals have been found in both meteorite and asteroid spectra, extrarrestrial goethite has never been identified within the meteorite inventory, so the possibility that putative goethite-containing asteroids never found a dynamical collisional pathway to Earth is less likely than the simpler interpretation: goethite is not present in asteroidal surfaces (Jewitt & Guilbert-Lepoutre 2012). Therefore, from the distribution of b-values, we conclude that most asteroids in this sample (which constitutes  ≲ 40% of the B-type population with computed R_p) present this absorption and that “water” (be it bound or free) may be common among the B-type asteroids.

4.3. The Pallas collisional family (PCF)

We use the most up-to-date Pallas family list by (Nesvorny 2012). WISE has observed 46 of the objects in this list. Histograms of beaming parameter and albedo determinations are shown in Fig. 10 (complete parameter set in Table 4). Given that the R_p value could only be fitted for seven objects, we do not include histograms for R_p and p_IR. In Table 6 we present the mean values of η, p_V, and R_p, along with their corresponding standard deviations and contributing number of determinations.

Fig. 7 b = 1 − Rp/R2.5 versus pV for a set of primitive control objects. The values of b obtained from the combination of our Rp and the R2.5 taken from the assembly of spectra published by other authors (cf. Fig. 6) and the shape of the 3 μm absorption band are consistent except for (13) Egeria.

Fig. 8 Normalised histograms showing the distribution of b-values. Top panel: B-types; the fact that b > 0 in the majority of cases suggests that a high percentage of these asteroids have absorption features in the  ~ 3-μm region. Bottom panel: S-types; the opposite conclusion is reached, consistent with the anhydrous nature of S-types.

The average η value of the PCF is consistent with that of the B-type population and that of the main belt (Masiero et al. 2011). However, the PCF has a moderately high albedo of  ~ 14%, significantly higher than the rest of the B-types and indeed higher than the value expected for primitive asteroids, whereas their average R_p is lower than the average value for the B-types (cf. Table 2). The seven R_p-values computed are quite homogeneous compared to the distribution observed for the rest of the B-type population. As we saw in Sect. 4.1, our parameter determinations are consistent within the error bars with those of Masiero et al. (2011), so we do not perform the same comparison for the PCF members. Masiero et al. show a histogram of log p_V with peak between log p_V = ( − 1) – ( − 0.8), i.e. p_V = 0.10–0.16, with which our results are consistent. (They provide no mean value to compare with.) On the other hand, though they provide the best-fit parameters in their Table 1, Masiero et al. do not discuss the R_p values of the PCF.

The average p_V and R_p values continue the trend observed for the clusters of de León et al. (2012) in Figs. 4 and 5: they have higher visible albedos on average than the G5 cluster and are bluer at 3.4 μm than the bluest cluster, G4. (The average R_p value of cluster G5 is obtained from only three objects, which could explain its deviation from the trend.) Members of the Pallas-like group of objects of Clark et al. (2010) were concentrated in clusters G4 and G5 in de León et al. (2012) (see Sect. 1). In this work, the list of Pallas family members observed by WISE includes still more asteroids. The four additional objects with R_p determinations that were not present in de León et al. (2012) are also located in the lower right-hand region of Fig. 4. This property would be consistent with the characteristics of an extrapolated G6 centroid (see the right panel of Fig. 11). Interestingly, the G6 centroid of de León et al. (2012) contained asteroid (3200) Phaethon alone. Therefore, the connection of the NEA (3200) Phaethon to the PCF established by de León et al. (2010) based on spectroscopical and dynamical arguments is also supported by the values of relative reflectances at 3.4 μm.

5. Discussion

In Sect. 4.2 we present statistically significant indications that water may be common within our sample, and in Sect. 4.3 we point out how the PCF have higher geometric visible albedos and more homogeneously distributed R_p-values than the rest of the B-types. The purpose of this section is to put these results in the context of other studies.

In the context of the geophysical models of the Themis and Pallas parent bodies by Castillo-Rogez & Schmidt (2010) and Schmidt & Castillo-Rogez (2012), the detections of water ice and organics on the surface of (24) Themis (Rivkin & Emery 2010; Campins et al. 2010b) and (65) Cybele (Licandro et al. 2011) have been invoked as provocative indications that water has played an important role in the accretion and evolution of these asteroids’ parent bodies in the mid-outer asteroid belt. Finding indications that a 3-μm is frequently present in B-types spectra adds further support to this conceptual framework, in which the gradual differences in the observed NIR spectral slopes of B-types might also be explained as resulting from different subsequent evolution and processing of their parent bodies.

Schmidt & Castillo-Rogez (2012) discuss that (1) Ceres, being the largest body of the asteroid belt and plausibly a water-rich asteroid, mostly preserved its integrity, whereas the Themis parent body was catastrophically disrupted. Pallas would be an intermediate case, still intact but showing evidence of heating, internal evolution, impact, and loss of water. In accordance with this scenario, the Themis family members included in the de León et al. (2012) sample were distributed more or less homogeneously among the representative centroids of the complete sample, spanning from neutral to red slopes and matching several meteorite analogues, whereas the PCF members were concentrated in the bluest centroids. As our results show, these trends are also verified longwards of 2.5 μm: the Themis family members show more heterogeneous R_p values, whereas the Pallas family members have distinctly higher albedos and lower, more homogeneous values of R_p than the rest of B-types. This homogeneity is consistent with the collisional family being the result of a non-disruptive cratering event on Pallas, which has a significantly higher albedo than the rest of large B-types.

Fig. 9 Schematic diagram illustrating the interpolation of R2.5 (empty points) from the average values of Rp (filled points) for the B and S asteroids studied.

Fig. 10 Beaming parameter (left) and geometric albedo (right) distributions of the values derived for the PCF members observed by WISE. The albedo distributions are normalised, i.e. divided by their respective total number of counts.

Fig. 11 Left panel: PCF members are on average brighter in the visible and bluer in the 3.4 μm region than the B-types sample (cf. Fig. 3). The 20% error bars are not plotted for clarity. Right panel: the average Rp of the PCF continues the decreasing trend of the de León et al. (2012) clusters (cf. Fig. 5).

The geophysical models by Schmidt & Castillo-Rogez (2012) favour a water-rich past for Pallas. Its near spherical shape may be explained as a consequence of early melting of a substantial initial icy component that is subsequently removed by a combination of thermal and impact processes. The water-loss processes that took place on the surface of Pallas may have taken place on other active bodies, such as main belt comets or indeed (3200) Phaethon, linked to the PCF by de León et al. (2010), and this activity has already been proposed to explain the characteristic NIR blue spectral slope (Schmidt & Castillo-Rogez 2012, and references therein). According to this model, water-rich materials are also expected to be important components within some members of Pallas’ family. The fact that the PCF members in this study are found to match the properties of the least hydrated clusters of de León et al. (2012) would be inconsistent with this view, but caution must be exercised when associating NIR slopes of meteorite analogues to hydration. For instance, while having suffered intense thermal metamorphism, CK4 chondrites (the best matches for Pallas) still show an absorption feature at 2.9 μm, albeit less prominent than e.g. CM chondrites.

From the dynamical standpoint, the simulations of Walsh et al. (2011) also provide a congruous context for our results. Their model suggests that C-complex asteroids were formed in the giant-planet-forming region and that, for every C-type planetesimal from beyond 8 AU that would later be located in the outer main belt, more than ten would have ended up in the region where terrestrial planets formed. Assuming that the composition of these objects is 10% water by mass, this may account for the minimal mass required to bring the current amount of water to our planet by a factor of 6–22.

On the other hand, the question of the moderately high values of p_V obtained for the PCF remains unanswered. Finding members of the Pallas and Themis collisional families distributed differently among the de León et al. (2012) clusters and in Fig. 11 (as discussed above) also leads us to ponder the possibility that a sequential or progressive physical process could explain the differences in the value distributions of p_V and R_p. Unfortunately, there are many possible mechanisms underlying the observed NIR spectral variability of primitive asteroids (e.g. regolith particle sizes, space weathering as a function of asteroid-family age and composition, thermal processing) and these are difficult to disentangle (Ziffer et al. 2011).

6. Conclusions

We have derived values of D, η, and R_p of 111 B-type asteroids by thermal modelling WISE data and updated H values (most of which have increased since the work of Masiero et al. 2011,as shown in Fig. 1). Additionally, from H and D we computed the corresponding values of p_V (Table 2). Our results agree within the error bars of the model with those previously published by Masiero et al. (2011). However, we obtained a systematic trend of approximately  − 10% discrepancies in the determinations of R_p that we cannot explain, but it is most probably attributable to small differences in the tabulated solar flux data needed to estimate the reflected light component at 3.4 μm. This work led to the following conclusions.

• 1.

  We derived the distribution of η, p_V, and R_P fot the B-type asteroids (see Fig. 2) and obtained the following mean values: , , and .

• There are no high-p_V, high-R_p B-type asteroids (p_V > 0.10, R_p > 1.0; see Fig. 3). The average R_p-values of the centroids of de León et al. (2012) clearly decrease from G1 to G5, which implies a relationship between the IR slope of the asteroid spectra up to 2.5 μm (Figs. 4 and 5).

• We computed b-values for a set of B-type asteroids and S-complex asteroids, which present and , respectively (Fig. 8). While the latter result is consistent with objects of the S-complex being anhydrous, the former indicates that the majority of B-type asteroids b-values in this study are consistent with the presence of a 3-μm absorption feature usually attributed to hydrated minerals or water ice; therefore, water must have played a key role in the evolution of a large fraction of the B-types, supporting recent works discussed in Sect. 5.

• We also studied the Pallas collisional family. On the one hand, the average albedo ( 0.14  ±  0.05) of this familly is significantly higher than the average albedo of B-types ( 0.07  ±  0.03) and moderately high compared to what is traditionally considered to be the albedo of primitive asteroids (<0.1). On the other hand, the albedo ratio values of the PCF members are very low and homogeneous ( 0.5  ±  0.1). These results clearly show the fundamental differences between this family and the rest of the B-types.

Our results support the scenarios by the geophysical models by Castillo-Rogez & Schmidt (2010) and Schmidt & Castillo-Rogez (2012) and the simulations of Walsh et al. (2011), which suggest that water played an important role in the origin of primitive asteroid parent bodies of the mid-outer belt. In addition, the connection of the NEA (3200) Phaethon to the PCF established by de León et al. (2010) based on spectroscopical and dynamical arguments is also supported by the values of relative reflectances at 3.4 μm.

Online material

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: ${\mathit{F}}_{\mathit{\lambda }}^{\mathrm{\left(}\mathit{m}\mathrm{\right)}}\mathrm{=}{\mathit{f}}_{\mathrm{th}\mathit{,\lambda }}\mathrm{+}{\mathit{r}}_{\mathrm{s}\mathit{,\lambda }}\mathit{.}$(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 $\frac{{\mathit{S}}_{\mathrm{\odot }}}{{\mathit{r}}_{\mathrm{\odot }}^{\mathrm{2}}}\mathrm{(}\mathrm{1}\mathrm{-}\mathit{A}\mathrm{)}{\mathit{\mu }}_{\mathit{i}}\mathit{\delta }{\mathit{a}}_{\mathit{i}}\mathrm{=}\mathit{\eta \sigma ϵ}{\mathit{T}}_{\mathit{i}}^{\mathrm{4}}\mathit{\delta }{\mathit{a}}_{\mathit{i}}\hspace{0.17em}\mathit{,}$(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 ${\mathit{f}}_{\mathrm{th}\mathit{,\lambda }}\mathrm{=}\mathrm{\Omega }\sum _{\mathit{i}}{\mathit{f}}_{\mathit{i,\lambda }}\mathrm{\left(}{\mathit{T}}_{\mathit{i}}\mathrm{\right)}\hspace{0.17em}\mathit{,}$(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: ${\mathit{r}}_{\mathrm{V}}\mathrm{=}{\mathit{f}}_{{\mathit{V}}_{\mathrm{\odot }}}\mathrm{\times }{\mathrm{10}}^{\mathrm{-}\frac{\mathit{V}\mathrm{-}{\mathit{V}}_{\mathrm{\odot }}}{\mathrm{2.5}}}\mathit{.}$(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. ${\mathit{r}}_{\mathit{\lambda }}\mathrm{=}{\mathit{r}}_{\mathrm{V}}\frac{{\mathit{B}}_{\mathit{\lambda }}\mathrm{\left(}{\mathit{T}}_{\mathrm{\odot }}\mathrm{\right)}}{{\mathit{B}}_{\mathrm{V}}\mathrm{\left(}{\mathit{T}}_{\mathrm{\odot }}\mathrm{\right)}}\mathrm{·}$(A.5)In this approximation, we can also consider $\frac{{\mathit{B}}_{\mathrm{IR}}\mathrm{\left(}{\mathit{T}}_{\mathrm{\odot }}\mathrm{\right)}}{{\mathit{B}}_{\mathrm{V}}\mathrm{\left(}{\mathit{T}}_{\mathrm{\odot }}\mathrm{\right)}}\mathrm{\approx }\frac{{\mathit{f}}_{\mathrm{I}{\mathrm{R}}_{\mathrm{\odot }}}}{{\mathit{f}}_{{\mathit{V}}_{\mathrm{\odot }}}}\mathit{,}$(A.6)from which we arrive at the following expression: ${\mathit{r}}_{\mathit{\lambda }}\mathrm{=}{\mathit{f}}_{\mathrm{I}{\mathrm{R}}_{\mathrm{\odot }}}\frac{{\mathit{B}}_{\mathit{\lambda }}\mathrm{\left(}{\mathit{T}}_{\mathrm{\odot }}\mathrm{\right)}}{{\mathit{B}}_{\mathrm{IR}}\mathrm{\left(}{\mathit{T}}_{\mathrm{\odot }}\mathrm{\right)}}\mathrm{\times }{\mathrm{10}}^{\mathrm{-}\frac{\mathit{V}\mathrm{-}{\mathit{V}}_{\mathrm{\odot }}}{\mathrm{2.5}}}\mathit{,}$(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 ${\mathit{r}}_{\mathrm{s}\mathit{,\lambda }}\mathrm{=}{\mathit{R}}_{\mathrm{p}}{\mathit{r}}_{\mathit{\lambda }}\mathit{.}$(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 ${\mathit{F}}_{\mathit{\lambda }}^{\mathrm{\left(}\mathit{m}\mathrm{\right)}}\mathrm{=}\mathrm{\Omega }\sum _{\mathit{i}}{\mathit{f}}_{\mathit{i,\lambda }}\left[{\mathit{T}}_{\mathit{i}}\mathrm{\left(}\mathit{\eta }\mathrm{\right)}\right]\mathrm{+}{\mathit{R}}_{\mathrm{p}}{\mathit{r}}_{\mathit{\lambda }}\mathit{.}$(A.9)We use the Levenberg-Marquardt algorithm (Press et al. 1986) in order to find the values of asteroid size ($\mathit{D}\mathrm{=}\sqrt{\mathrm{\Omega }}$, in km), beaming parameter (η) and albedo ratio (R_p) that minimise the χ² of the asteroid’s WISE data set, namely ${\mathit{\chi }}^{\mathrm{2}}\mathrm{=}\sum _{\mathit{j,\lambda }}{\left(\frac{{\mathit{F}}_{\mathit{j,\lambda }}\mathrm{-}{\mathit{F}}_{\mathit{j,\lambda }}^{\mathrm{\left(}\mathit{m}\mathrm{\right)}}}{{\mathit{\sigma }}_{\mathit{j,\lambda }}}\right)}^{\mathrm{2}}\mathit{,}$(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 ${\mathit{F}}_{\mathit{\lambda }}^{\mathrm{\left(}\mathit{m}\mathrm{\right)}}$ 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 $\frac{\mathit{\partial }{\mathit{F}}_{\mathit{\lambda }}^{\mathrm{\left(}\mathit{m}\mathrm{\right)}}}{\mathit{\partial \eta }}\mathrm{=}\left(\frac{\mathit{\partial }{\mathit{F}}_{\mathit{\lambda }}^{\mathrm{\left(}\mathit{m}\mathrm{\right)}}}{\mathit{\partial }{\mathit{T}}_{\mathit{i}}}\right)\left(\frac{\mathit{\partial }{\mathit{T}}_{\mathit{i}}}{\mathit{\partial \eta }}\right)\mathrm{·}$(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.

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).

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.

Acknowledgments

We thank the referee for a careful and constructive revision. V.A.L. acknowledges support from the project AYA2011-29489-C03-02 (MEC). J.L. acknowledges support from the projects AYA2011-29489-C03-02 and AYA2012-39115-C03-03 (MINECO). J.d.L. is grateful for financial support via a “Juan de la Cierva” contract from the Spanish “Secretaría de Estado de Investigación, Desarrollo e Innovación”. M.D.B. thanks the Space Situation Awareness programme of the European Space Agency (ESA-SSA) for financial support. N.P.A. was financed by the Spanish Ministry of Economy and Competitiveness through the “Juan de la Cierva” programme. This publication makes use of data products from NEOWISE, which is a project of the Jet Propulsion Laboratory/California Institute of Technology, funded by the Planetary Science Division of the National Aeronautics and Space Administration. This research also made use of the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.

References

All Tables

All Figures

	Fig. 1 Differences in absolute magnitude values between those used by Masiero et al. (2011), HM, and the most updated ones (as of May 2012) used in this paper, HU. Note that the cases verifying ΔH = 0 do not contribute to this histogram.
In the text

	Fig. 2 Histograms of value distributions obtained for the B-type asteroids observed by WISE: η (top left), pV (bottom left), Rp (top right), and pIR (bottom right). The corresponding histograms using the results from Masiero et al. (2011) are overplotted.
In the text

	Fig. 3 Albedo ratio versus visible geometric albedo. The 20% error bars in pV and Rp are not shown to facilitate visualisation. B-types observed by WISE are plotted in black circles; all main belt objects featured in Table 1 of Masiero et al. (2011) are in empty grey circles. The horizontal line is an artificial feature corresponding to objects with the default fixed value of Rp = 1.5.
In the text

	Fig. 4 Rp vs. pV for the asteroids in de León et al. (2012) distinguishing the clusters to which they belong. Within parentheses, the average pV of each cluster is shown. On average, pV increases from G1 to G3, keeping the same value from G3 to G5.
In the text

	Fig. 5 Average values of Rp for each cluster defined by de León et al. (2012). In terms of spectral slope up to 2.5 μm, cluster G5 is the bluest, whereas G1 is the reddest. The error bar is the standard error of the mean.
In the text

	Fig. 6 Assembled spectra of the control objects chosen to study the reflectivity at 3.4 μm derived from WISE data as a diagnostic of the presence of the 3-μm feature. The dashed lines approximately enclose the wavelength integration range of the W1 filter.
In the text

	Fig. 7 b = 1 − Rp/R2.5 versus pV for a set of primitive control objects. The values of b obtained from the combination of our Rp and the R2.5 taken from the assembly of spectra published by other authors (cf. Fig. 6) and the shape of the 3 μm absorption band are consistent except for (13) Egeria.
In the text

	Fig. 8 Normalised histograms showing the distribution of b-values. Top panel: B-types; the fact that b > 0 in the majority of cases suggests that a high percentage of these asteroids have absorption features in the  ~ 3-μm region. Bottom panel: S-types; the opposite conclusion is reached, consistent with the anhydrous nature of S-types.
In the text

	Fig. 9 Schematic diagram illustrating the interpolation of R2.5 (empty points) from the average values of Rp (filled points) for the B and S asteroids studied.
In the text

	Fig. 10 Beaming parameter (left) and geometric albedo (right) distributions of the values derived for the PCF members observed by WISE. The albedo distributions are normalised, i.e. divided by their respective total number of counts.
In the text

	Fig. 11 Left panel: PCF members are on average brighter in the visible and bluer in the 3.4 μm region than the B-types sample (cf. Fig. 3). The 20% error bars are not plotted for clarity. Right panel: the average Rp of the PCF continues the decreasing trend of the de León et al. (2012) clusters (cf. Fig. 5).
In the text

	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.
In the text

	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).
In the text