© The Authors 2023

1 Introduction

As remnants of the processes involved in planet formation, asteroids make up a negligible portion of the total mass of our solar system. Nonetheless, their abundance and diversity, as seen in observations, provide a fascinating glimpse into the intricacies of the early evolution of planetesimals. Among them, so-called C-complex asteroids with low albedo (≲0.1; Bowell et al. 1978; Tholen 1984) have been garnering steady attention due to their volatile-rich (icy) composition (Rivkin et al. 2015a), which might have an impact on habitability (Kwok 2018). The presence of volatile materials (or traces of their past existence) makes C-complex asteroids useful for linking observations to their early history.

C-complex asteroids, which account for around half of the asteroid belt clustering between Mars and Jupiter (DeMeo & Carry 2014), are classified into several taxonomic types (Tholen 1984; DeMeo et al. 2009) based on the properties of their reflectance spectra at visible and near-infrared wavelengths (0.4– 2.5 µm; VNIR). Variations in the VNIR spectral properties can be attributed to both the inherent compositional and physical properties of grains on asteroidal surfaces (i.e., regoliths). Composition-wise, relatively featureless spectra with negative-to-moderately positive VNIR slopes indicate that they are historically connected to carbonaceous meteorites with varying contents of carbons, metals, and silicates (Johnson & Fanale 1973). Post-accretionary processes would alter the natal composition of the asteroids. In particular, aqueous alteration, a low-temperature process where melting ice interacts with anhydrous silicates and metals, has been observed across various subclasses of C-complex asteroids. This process produces unique absorption features, such as phyllosilicate bands near 0.7 and 3 µm (e.g., Fornasier et al. 2014; Rivkin et al. 2022), and/or changes spectral slopes in the VNIR (Trang et al. 2021). The physical makeup of regoliths, such as size and texture, is strongly dependent on the collisional history of the bodies whose characteristic lifetime is a function of the diameter (Bottke et al. 2005). While large (≳ 100 km) asteroids are regarded as primordial objects that have well-developed particulate regoliths over a sufficient timescale, smaller (< 100 km) asteroids have undergone active surface rejuvenation and contain particles that are relatively larger in size due to the weak gravitational force of the bodies (Dollfus et al. 1989; Delbo et al. 2007; Marchis et al. 2012).

Polarimetry has long been used to study regolith environments. Asteroids scatter light in a manner that encodes details about their surfaces by interacting electromagnetically with the regolith particles. Upon scattering, the light becomes partially polarized, where the degree of polarization and its variations with phase angle α (angle of the Sun–asteroid–observer) and wavelength provide a useful diagnostics of the surface nature (Cellino et al. 2015b). In particular, the reversal of polarization position angles around the backscattering region (α ≲ 30° ) producing the negative polarization branch (NPB; Shkuratov 1985) is one of the hallmarks of asteroids’ regoliths. Instead of simple Fresnel’s reflection, complicated interferences between multiple-scattered light by regolith particles (coherent backscattering mechanism; Muinonen et al. 2015 and references therein) should be taken into account. Given that the shape of the NPB ties to compositional and textural information of regoliths, polarimetry has been successfully applied to lunar regolith studies (Dollfus & Geake 1975, 1998; Shkuratov 1992), albedo estimations of asteroids (Zellner et al. 1977a; Geake & Dollfus 1986; Cellino et al. 2015a), and taxonomy studies (Belskaya et al. 2017; Masiero et al. 2022). Spectropolarimetry has emerged in recent years as a promising technique for exploring asteroid regoliths. Bagnulo et al. (2015) showed that asteroids with similar reflectance spectra may exhibit significantly different polarization spectra, yet there have been no follow-up studies; this is primarily because polarimeters are more difficult to access than traditional imaging or spectroscopic facilities.

We here present new spectropolarimetric observations of large C-complex asteroids over 420–930 nm from the European Southern Observatory’s Very Large Telescope (ESO/VLT) and the William Herschel Telescope (WHT). In this paper, we focus on large (≳100 km) low-albedo (≲0.1) C-complex asteroids, including some that have not yet been fully explored from a polarimetric perspective. The adopted size criterion of the asteroids would help reduce the potential impact of different regolith sizes on optical behavior and secure strong enough signals to obtain accurate measurements of small amounts of polarization (typical maximum polarization in the NPB for asteroids is ≳2%; Cellino et al. 2015b). We evaluate the spectropolarimetric properties of our 12 targets (whose orbital distributions are shown in Fig. 1) and the correlation between the shapes of their polarization and reflectance spectra. With the aid of 52 additional C-complex asteroids acquired through polarimetric surveys, we examine the distribution of the inversion angle (defining the angular width of the NPB) and its relationship to the morphology of the reflectance spectra for a total of 64 asteroids belonging to different subclasses.

The restricted range in size and taxonomy of the objects considered in our analysis provides only a partial picture of the overall asteroid population. However, by presenting the first systematic polarimetric study of these low-albedo asteroid groups and showing how their polarimetric behavior presumably correlates with reflectance attributes, this study is intended to help ascribe the primary surface characteristics of asteroids influencing optical wavelength observations.

2 Observations and data analysis

Spectropolarimetric observations of 12 C-complex asteroids were conducted over 2014–2016 using the FORS2 instrument mounted on the Cassegrain focus of the 8.0-m diameter UT1 telescope of the VLT at the Paranal Observatory in Chile. FORS2, FOcal Reducer and low-dispersion Spectrograph, is a versatile instrument that offers multi-object spectropolarimetry (PMOS) mode. The same instrumental setup was applied throughout all PMOS observations, using grism 300V with no order separating filter, and a 2″ slit width, for a spectral coverage between 400 and 930 nm, and a resolving power of ~220 (more details are given in Bagnulo et al. 2015). Two asteroids were observed with the ISIS instrument of the WHT. Details on the instrument and observing technique are given in Bagnulo & Landstreet (2019). ISIS observations were obtained using grating R300B in the instrument blue arm (range 370–590 nm) and grating R158R plus order-sorting filter GG495 in the red arm (range 480–975 nm). A journal of observations is given in Table 1.

Raw images were preprocessed using standard methods (bias subtraction, cosmic-ray removal, and sky subtraction). The calibration and extraction of polarimetric parameters were conducted in the same way as in Bagnulo et al. (2015): the observed Stokes profiles were rotated to take into account the chromatism of the retarder waveplate. This is explained, for instance, in Sect. 4.1 of Bagnulo & Landstreet (2019)¹. As is customary in solar system science, the corrected degree of linear polarization, P, and its position angle, θ_P, (with regard to the equatorial system; Eqs. (6) and (7) in Kwon et al. 2022a) were transformed into a system whose reference direction is perpendicular to the scattering plane (a plane including the Sun-asteroid-observer) using

(1)

where

(2)

in the same manner as in Chernova et al. (1993). Here, ϕ is the position angle of the scattering plane. Input ϕ was adjusted to satisfy 0 ≤ (ϕ ± π/2) ≤ π. The reduced (rotated) P_r value now has either a negative or positive sign depending on viewing geometry, that is, P_r > 0 is the polarization perpendicular to the scattering plane and P_r < 0 is the polarization parallel to the scattering plane. Accordingly, θ_r should be ideally 0° in the positive polarization branch (PPB) and 90° in the negative polarization branch (NPB), which serves as an independent check for the reliability of our data reduction procedures. The polarization spectra P_r(λ) over 420–930 nm for our 12 C-complex asteroids are shown in Fig. 2, along with the spectral distribution of θ_r All θ_r values cluster around either 0° or 90° as expected, except at the short and long ends of the wavelength considered, where grism efficiencies are lower and at the wavelengths of telluric absorption bands (e.g., O₂ absorption at ~760 nm).

Fig. 1 Orbital distributions of 12 C-complex asteroids observed with VLT/FORS2 in spectropolarimetry Background grey dots are numbered main-belt asteroids that are quoted from JPL Small-Body Database Query. Symbols in the highlighted color are proportional to the size of asteroids. Vertical lines at different heliocentric distances indicate the most important mean-motion resonances with Jupiter.

3 Results

This section examines the wavelength dependence of polarization P_r(λ) for our 12 C-complex target asteroids. With 52 more asteroids having V-band polarization data from polarimetric surveys, we analyze the similarities and differences between their phase-angle dependence P_r(α).

3.1 Polarimetric dependence on wavelength

Figure 2 presents reduced polarization P_r and its position angle θ_r (Eqs. (1) and (2)) of our target asteroids as a function of wavelength. Each panel has a different y-axis scale. We rebinned data at ~3 nm intervals. The lowest-resolution data (Δλ ~ 20– 40 nm) is highlighted with salmon-colored diamond symbols. The sharp deviations of P_r and θ_r around telluric absorption regions in the finely-binned data are absent in the coarsest data. Also, P_r exhibits neither abrupt absorption nor emission features except near both ends of the wavelength regions and rather varies smoothly. The overall slope of P_r(λ) in the NPB is either neutral to negative, which is different even for some asteroids in the same taxonomic type that have similar VNIR reflectance spectra. There is no apparent correlation between the slope and the phase angle of observations. Interestingly, two epochs (C-type (35) Leukothea at α =23°4 and Ch-type (404) Arsinoe at α = 23°4) in the PPB have a moderately positive slope; also, one of T-type (308) Polyxo at α = 20°.8 appears to be at the very beginning of a slope reversal, in contrast to their trends in the NPB. A similar transition – from a negative-to-neutral gradient in the NPB to a slightly positive gradient in the PPB – has also been observed for intermediate-albedo asteroids (Bagnulo et al. 2015). We focus on the observations in the NPB throughout the study to characterize the scattering environment of asteroids dominated by coherent backscattering; however, observations over a broader α (covering both NPB and PPB) should reveal differences and similarities among the asteroids in greater detail, which will be the topic of future work.

To compare the shape of polarization spectra, we took absolute values of each spectrum |P_r(λ)| and normalized the curve at 550 nm. However, the direct comparison of asteroids observed at different α should be done with caution. An asteroid’s polarization spectrum would not actually reverse its slope (from positive to negative or vice versa) and, in fact, would likely be maintained in the NPB, such as (1) Ceres (see Fig. 2a) and some asteroids discussed in Bagnulo et al. (2015), although a slight slope variation may occur as α changes. For instance, one epoch of (308) Polyxo at α =20°8 near the inversion angle exhibits a neutral slope (see Fig. 2i) compared to the trend observed at α =15.8. The slope’s turn to naught as the NPB nears the inversion angle may be a natural consequence of the general shape of the asteroids’ NPB: the NPB has a bowl shape with a minimum polarization degree P_min (i.e., maximum |P_r|) at α_min ~ 10° and smaller |P_r| values at other phases with zero at the inversion angle (Cellino et al. 2015b). As such, lumping together asteroids of different α facilitates qualitative analysis of trends in polarization spectrum slopes, but their quantitative differences would be only temporary. This is in agreement with the assumption of Bagnulo et al. (2015) that the α difference between the asteroids may not significantly affect (at least to the first order) the interpretation of the normalized polarimetric spectra in the NPB.

We excluded the data taken around the inversion angle because of their neutral slope with near-zero P_r (e.g. (308) Polyxo at α = 20°.8 and (444) Gyptis at α = 21°.2) and data of (175) Andromache due to its wavy structure that is hardly interpretable in the normalized spectrum. As a result, we sorted out 11 normalized spectra (one epoch for each asteroid) for comparison. A quadratic function was used to fit the moderately binned (Δλ ~ 3–5.2 nm) |P_r| spectra over 450–900 nm. For asteroids showing a strong feature around both ends of the wavelength (probably an artifact associated with low signals), in particular, (128) Nemesis whose errors exceed 50% at wavelengths beyond 750 nm (panel g in Fig. 2), we only examined data between 450 and 750 nm. Figure 3 illustrates the modeled curves for each asteroid over 420–930 nm based on their best-fitting solutions.

Here, we define a “polarization color” as the slope of a polarization spectrum and label it as blue (red) when the absolute P_r decreases (increases) as wavelength increases. Accordingly, an asteroid showing a positive (negative) slope in Fig. 2 has a blue (red) polarization color. In Fig. 3, the asteroids display a (quasi-)linear slope whose color ranges from gray (0.01 ±0.01% per 100 nm) to blue (−0.12±0.01% per 100 nm) over 500– 800 nm, though most vary within ±5% of the normalized point at 550 nm. (10) Hygiea and (451) Patientia stand out for their blue polarization colors varying larger than 20% over the entire wavelength considered, whilst those of four outer-belt asteroids – (24) Themis, (90) Antiope, (120) Lachesis, and (128) Nemesis – is hardly distinguishable.

Three asteroids (given separately in an inserted window in Fig. 3) have somewhat concave shapes with a local |P_r | minimum between 600 and 720 nm: T-type (308) Polyxo, Ch-type (404) Arsinoe, and C-type (444) Gyptis in descending order of concavity. The concavity is always present, regardless of the spectral resolution of the data used for fitting, and leads to a change in slope, creating stronger polarization in the red part of the wavelength. The wavelength position of the local |P_r | minimum for (308) Polyxo appears consistently at a shorter wavelength around 600 nm when compared to those of the other two asteroids around 700–720 nm. It is interesting to note that this spectral region (the so-called 0.7-µm region; Rivkin et al. 2002) has been widely employed as a diagnostic for a Fe²⁺−Fe³⁺ charge-transfer band of phyllosilicates, a product of aqueous alteration (Vilas & Gaffey 1989; Vilas 1994). The three asteroids with less polarized signals in this region have indeed in common that their reflectance spectra contain a measurable 0.7-µm band but also a sharp band at ~2.7–2.9 µm, making them belong to the phyllosilicate-dominant “sharp” group (Takir & Emery 2012). Furthermore, the fact that large T-type asteroids, including (308) Polyxo, possess a 0.7-µm band centered around 600–650 nm (Kwon et al. 2022b), namely, a shorter side than that of ordinary aqueous-altered asteroids centered around 700–750 nm (Fornasier et al. 2014; Rivkin et al. 2015b), seems aptly compatible with the observed difference in the local |P_r | minimum wavelength of the three asteroids given in Fig. 3. The asteroids displaying linear polarization spectra are devoid of a corresponding infrared absorption feature and belong to the “non-sharp” group (Rivkin et al. 2022 and references therein).

Systematic differences in the porosity (or rugosity) of the regoliths of the asteroids could affect the observed diversity in the linear slope of their polarization spectra. As dust porosity determines the effective size of working units, regoliths of different porosities will exhibit a difference in light scattering cross-sections (Kolokolova 2011). In the case of large low-albedo asteroids, fine-grained (~1–10 µm) regoliths tend to exhibit more red (positive) spectral slopes in reflectance spectra than larger (≳10–100 µm) regoliths (e.g. Emery 2022; Emery & Brown 2004). The observed variation in slope may correspond to variations in the regolith environment that are distributed across a range of sizes if the amount of scattered light from the surface (albedo) is linearly related to the degree of polarization at a given wavelength on our sample of asteroids.

We find no correlation between the shape of polarization spectra (in terms of negativity and curvature) and phase angles, diameters, albedos, and semimajor axes of the asteroids, whose most widely accepted values are summarized in Table 2. The optical spectropolarimetric properties of the large C-complex asteroids are therefore likely to be primarily determined by their surface composition, which is in line with recent spectropo-larimetric observations of Didymos-Dimorphos binary systems conducted throughout the DART impact (Bagnulo et al. 2023).

Fig. 2 Reduced polarization degree, Pr, and its position vector, θr, as a function of wavelength for 12 C-complex asteroids taken at 17 epochs (Table 1). In each panel, the phase angle (α) at the time of observation is provided in the legend. Different colors were used for asteroids observed more than once. The curve with salmon-colored diamonds represents the binned polarization spectrum with the coarsest spectral resolution (an interval of ~20–40 nm).

Fig. 3 Best-fitting quadratic curves of polarization spectra of the target asteroids normalized at 550 nm. Three asteroids showing concave shapes – (308) Polyxo, (404) Arsinoe, and (444) Gyptis - are visualized in a separate window.

3.2 Polarimetric dependence on phase angle

Asteroids draw a unique bell-shaped polarimetric-phase curve P_r(α) that can be described as follows: as a phase angle α increases from zero, P_r becomes more negative, reaches its minimum P_min at α_min of ~5–15°, and then rebounds to zero P_r at the inversion angle α₀ of ~15–30° beyond which P_r becomes positive (Cellino et al. 2015b; Belskaya et al. 2017). Two chief parameters defining the shape of the NPB – α₀ (the angular width of the NPB) and P_min (the depth of the NPB, usually in absolute number) – are of particular importance thanks to their sensitivity to the regolith’s compositional and physical properties (Dollfus & Geake 1975; Geake & Dollfus 1986; Belskaya 2005).

To grasp the P_r(α) of our C-complex asteroids in the first place, we averaged the polarization data bracketed in the standard B (405–485 nm), V (510–590 nm), R (620–670 nm), and I (730–880 nm) band regions. Table 3 summarizes the weighted-average P_r,X and its error in each wavelength domain, X. As for the B and I bands, the presence of deviating structures in polarization spectra as signals weaken (Fig. 2) makes the data less reliable than the values of the V and R bands. As archival V-band polarimetric data significantly outnumber those of other bands, we utilized V-band data for the analysis. Figure 4 shows the polarization-phase curves of our targets in the V band. Data points from this study (Table 3) are highlighted as gray-scale symbols (white and black indicate the brightest and darkest ends of (175) Andromache and (404) Arsinoe, respectively), while the background circles represent their archival data quoted from the NASA Planetary Data System (PDS) Small Bodies Node (Lupishko 2022) and Calern Asteroid Polarization Survey (CAPS; Bendjoya et al. 2022).

The P_r,_V values from our spectropolarimetric results share a consistent phase-angle dependence with the existing datasets derived from aperture polarimetry. Despite the different subclasses, the C-complex asteroids draw a broadly homogeneous P_r(α), as shown in Belskaya et al. (2017). The data points show no obvious correlation between the asteroids’ albedo and relative P_r excess at the given a, although, in general, the more diffuse light component contributes to the observed scattered light (i.e., higher albedo surfaces), the easier it is to lose the directional information and become depolarized. A lack of such an inverse relationship between albedo and polarization (the so-called Umov law; Umov 1905) in our C-complex asteroids (or at least in the NPB we are exploring) is consistent with previous laboratory studies showing that the Umov law becomes saturated at an albedo of ~0.08 (Zellner et al. 1977a,b; Geake & Dollfus 1986; Cellino et al. 2015a).

Next, we increased the number of large C-complex asteroids from NASA/PDS and CAPS archives in order to parameterize the NPB phase curve using a conventional trigonometric function (Lumme & Muninonen 1993):

(3)

or through the linear-exponential function (Kaasalainen et al. 2003):

(4)

where constants b, c₁, c₁, and α₀ (inversion angle) in Eq. (3) and A, B, and C in Eq. (4) are free parameters shaping the curve, retrieved via best-fit techniques.

With these two fitting functions in hand, we only considered asteroids if they simultaneously satisfy the following criteria. First, the asteroids considered are large (D_e > 100 km) C-complex asteroids. Second, they should have a minimum of four V-band measurements including the point (0,0), that is, P_r = 0% at a = 0°, assuming that there is no significant alignment of regolith particles. Third, their observations should be evenly distributed over α covering more than 10° in the NPB. Fourth, their α₀ and P_min values retrieved from the trigonometric and linear-exponential fitting functions should be agreed within the error bars. Lastly, a fitting uncertainty should be <20%.

Condition 4 arises from the empirical nature of both fitting functions which makes judging one over the other pointless. This constraint is usually met if there are sufficient observations available. When compared to Eq. (3), which is capable of fitting polarization-phase curves up to high phases, Eq. (4) generally fits the curves of main-belt asteroids whose α-coverage hardly exceeds 30°. The final estimates were selected based upon the conventional trigonometric function (Eq. (3)), though we are dealing with main-belt asteroids. This is because within the limit of 5000 iterations we arbitrarily set, Eq. (3) tends to provide more stable error estimates than Eq. (4); this is possibly due to a lack of dense coverage at small α, as pointed by Kaasalainen et al. (2003). Consequently, 64 asteroids were selected, whose retrieved polarimetric parameters with 1α errors are available in Table 2 along with their taxonomic types. In some cases, C-complex asteroids have more than one subclass or are classified differently under Tholen (Tholen 1984) and Bus-DeMeo systems (SMASSII; Bus & Binzel 2002; DeMeo et al. 2009). In Table 2, the selected taxonomic type is highlighted in boldface and the reasons for selection are given in Appendix A.

Figure 5 summarizes the fitting results. A pair of border lines measured from lunar samples (Geake & Dollfus 1986; Dollfus et al. 1989) separate regolith regimes into three domains based on α₀ values: bare-rock/bouldery (leftmost), particulate (intermediate), and fine-dust and/or fluffy (rightmost) regoliths. The |P_min| range of the asteroids is consistent with that of carbonaceous meteorites (Zellner et al. 1977a,b). Figures 5a,b display the distribution of asteroids having different taxonomic types in the retrieved α₀–|P_min| space. The distribution of asteroids (F, B, T, and Ch types) in panel a illustrates the localization of each subgroup within the parameter space, while in panel b, taxonomic types are either distributed throughout the entire parameter range (P and C/Xc types) or are lacking sufficient data to draw any meaningful conclusions (Cg and Cb types).

The α₀ and |P_min| of F-, B-, T-, and Ch-type asteroids in Fig. 5a are proportional, and the stream aligns with the direction of positive correlation (i.e., the stronger the negative polarization, the wider the NPB) in ascending order of F (weighted mean and its 1-σ error: α₀ = 17° 10 ± 1°64 and |P_min| = 1.22 ± 0.09%), B (α₀ = 19°04 ± 0°48 and |P_min| = 1.40 ± 0.16%), T (α₀ = 21°14 ± 0°14 and |P_min| =1.37 ± 0.01%), and Ch (α₀ =21°07 ± 0°47 and |P_min| = 1.76 ± 0.19%) types. The Spearman correlation coefficient ρ and its p-value of the distribution as a whole are and p = 0.0001, respectively. The statistical details are provided in Appendix B. F-type asteroids occupy the smaller end of α₀ compared to most of the C-complex asteroids, as noted in previous studies (Belskaya 2005; Belskaya et al. 2017).

It is intriguing to see that the sequence of distribution (F → B → T → Ch) in the parameter space appears to parallel that of the expected degree of aqueous alteration (P < B < T ≲ Ch, no F types in the samples) based on the detection rate of a 0.7-µm band and the depth of a 2.9-µm band depth (Fornasier et al. 2014; Kwon et al. 2022b). About half of the F- and Ch-type points reside outside the particulate domain where most large asteroids are expected to be positioned (Marchis et al. 2012). We return to these topics in greater detail in Sect. 4.

C/Xc-type asteroids distribute across the entire particulate domain (Fig. 5b). This apparent diversity indicates that this taxonomic group may indeed have highly heterogeneous surfaces and/or they may belong to another C-complex subgroup, but their diagnostic features may be too weak to be detected.

Lastly, we find that there is a weak but notable correlation between α₀ and the diameter of asteroids whose NPB parameters are within the particulate domain (Fig. 5c). The asteroids analyzed here roughly align with the inverse α₀–diameter correlation ( and p = 0.01; Appendix B) as a whole and possibly within each taxonomic group: most notably in C-type asteroids ( and p = 0.04). According to the α₀–regolith size relationship (assuming that this bordering scheme is applicable to large C-type asteroids), the larger asteroids with smaller α₀ values would have coarser regolith grains than the smaller asteroids with larger α₀ values. This this correlation. Aside from the results shown in Fig. 5, we are unable to find statistically significant relationships between the polarimetric parameters and the surface characteristics (e.g., albedo) of the asteroids.

Fig. 4 V-band polarization-phase curve Pr(α) of our 12 target asteroids. Symbols with thick line edges represent the data measured from this study, with gray-scale colors indicating their relative albedos ranging from white (high-albedo end: 0.094 of (175) Andromache) and black (low-albedo end: 0.037 of (404) Arsinoe). The albedos were from the NASA JPL Small-Body Database. Background circles are their archival data from the NASA Planetary Data System (PDS) Small Bodies Node (Lupishko 2022) and Calern Asteroid Polarization Survey (CAPS; Bendjoya et al. 2022). Points of the dwarf planet (1) Ceres are distinguished by black-edged orange circles.

Fig. 5 Distribution of 64 large C-complex asteroids in the retrieved inversion angle α0 and absolute polarization minimum |Pmin| space (panels a and b), along with the size of the asteroids (panel c). Guidelines in panels a and b represent the borders of surface regimes dominated by different-size regolith particles (Geake & Dollfus 1986): the left border separates the boulder and bare-rock regoliths (leftmost) and particular surface domain (intermediate) and the right one divides the particulate domain and fine-dust regoliths (rightmost). Different colors are assigned to different taxonomic types. Panel c illustrates the distribution of asteroids in the particulate domain.

4 Discussion

This section provides polarimetric evidence for surface heterogeneity in large C-complex asteroids. Using the observed correlation of polarimetric and spectroscopic properties, we attempt to tie VNIR observations to asteroids’ surface environments and discuss what future polarimetric research might be most useful.

4.1 Correlation between polarization and reflectance spectra

Electromagnetic waves have three fundamental properties: intensity, frequency, and polarization. As each of them has a different sensitivity to a specific characteristic of scatterers, combining the information from more than one probe provides excellent complementary information for constraining the scattering environment (Bohren & Huffman 1983). The polarization spectra P_r(λ) of large C-complex asteroids (Fig. 3) have already revealed some heterogeneity within a limited sample, motivating us to explore their correlation with reflectance spectra I (λ).

To search for the correlation between P_r(λ) and I(λ) for our target asteroids, we employed archival reflectance spectra from the second phase of the Small Main-belt Asteroid Spectroscopic Survey (SMASSII; Bus & Binzel 2002; DeMeo et al. 2009) and the Small Solar System Objects Spectroscopic Survey (S³OS²; Lazzaro et al. 2004). Figure C.1 displays I(λ) of the asteroids at the visible wavelength (~450–950 nm). The archives provide normalized reflectance, which is an observed spectrum divided by that of a solar analog, which is in turn normalized at 550 nm. In theory, a reflectance spectrum may be constructed from our polarimetric observations; however, we did not obtain observations of solar analogs in a systematic manner: only a couple of solar analogs were observed in the course of the entire semester of observations and thus they do not necessarily have a similar airmass to the asteroids’. The mismatch in sky transparency can result in a discrepancy in the shape of the synthesized spectrum versus the existing reflectance data; thus, we decided to use the archival data. Although a difference in phase angle between archival and our spectra may affect a slope correlation analysis, the phase-reddening effect (variation of the spectral slope with phase angle; Gradie & Veverka 1986) has been shown to be insignificant for asteroids analogous to our targets over a phase angle range of 0–30° (Beck et al. 2021). We confirmed that for spectra available for multiple epochs for a single asteroid (Fig. C.1), the spectral shape is consistent within the measurement error; that is, except for (145) Adeona, whose discrepancy seems systematic rather than phase-related. We thus assumed that phase-reddening effects on the asteroids in our sample are negligible and used average data points. The absolute polarization degree, |P_r(λ)|, and reflectance, I(λ), were interpolated to match spectral resolution (450 to 900 nm with a 5 nm step) and normalized at 550 nm. To investigate the variation of |P_r(λ)| with a change in the shape of I(λ), we weight-averaged the |P_r(λ)| values in each spectral bin (450–550 nm, 550–650 nm, 650–750 nm, and 750–900 nm) and measured a spectral slope, S, in units of% (100 nm)⁻¹ in a given range.

The distribution of the large C-complex asteroids in the |P_r(λ)|–S space is illustrated in Fig. 6. Circles and diamonds indicate the short- and long-ends of the spectral bins (450–550 nm and 750–900 nm, respectively). A single line for an asteroid connects four points, each of which represents the normalized |P_r| and S values with a 1σ standard deviation per spectral bin. At a glance, the majority of curves (colored in green) cluster within ±5% of the horizontal line of unity, indicating a relatively neutral slope of their polarization spectra. Three asteroids showing appreciable variations are highlighted – C-type (10) Hygiea, T-type (308) Polyxo, and C-type (451) Patientia – and their ways of correlating |P_r| and S in the most part of the optical wavelength is twofold: the asteroids are located in either the first (upper right) or the third (lower left) quadrant in which |P_r(λ)| and S are positively correlated. In other words, lower (higher) |P_r| values with regard to the normalization point at 550 nm are associated with negative (positive) spectral slopes. This is in contrast to siliceous S-complex asteroids with moderate-to-high albedos (>0.1) that obey the Umov law, thereby occupying the second and fourth quadrants (Bagnulo et al. 2015, 2023).

Interestingly, the asteroids showing dichotomic trends here correspond to the asteroids in a distinct shape of polarization spectrum (Fig. 3): (10) Hygiea and (451) Patientia stand out for their steep blue polarization color, whilst (308) Polyxo has the largest concavity in its polarization spectrum. In Fig. 7, we examine separately the dependence of asteroids with convex polarization spectra such as (308) Polyxo. Their stretch in parameter space is proportional to the concavity of the polarization spectrum shown in Fig. 3. Near 550–750 nm, (308) Polyxo and (404) Arsinoe exhibit a local V shape whose vertex appears around a 0.7-µm band center on their reflectance spectra. This indicates the presence of materials with different optical properties than those of the background components. (444) Gyptis, on the other hand, has the weakest concavity of the three and resembles the shape of other C-complex asteroids in Fig. 6.

The asteroids exhibiting a dichotomy in the |P_r(λ)|-S correlation have the largest difference in α₀ (either largest or smallest) values among the asteroids analyzed in Fig. 6. In Fig. 5a, the smaller-α₀ part is predominantly occupied by F- and B-type asteroids with linear, featureless reflectance spectra in a negative-to-neutral spectral slope (Gradie & Tedescro 1982; Bus & Binzel 2002), while the larger-α₀ part is mostly occupied by Ch- and T-type asteroids having characteristic absorption bands in the 0.7-µm region in a neutral-to-moderately-positive color (Rivkin et al. 2015b). The finding that asteroids with a large difference in α₀ also exhibit noticeable differences in the |P_r(λ)|– S space may suggest the presence of a common mechanism underlying the optical phenomena of large C-complex asteroids.

Fig. 6 Correlation between the spectra of the normalized absolute polarization degree |Pr(λ)| and a spectral slope S of reflectance spectra for the C-complex asteroids in Fig. 3. Both parameters were normalized at 550 nm and measured in each spectral bin. Circles and diamonds of each curve indicate the values derived in the shortest (450–550 nm) and longest (750–900 nm) bins, respectively. Including those endpoints, a single line connecting a total of four points represent |Pr| and S values of an asteroid evaluated over 450–550 nm, 550–650 nm, 650–750 nm, and 750–900 nm. Only three asteroids – (10) Hygiea, (308) Polyxo, and (451) Patientia – are highlighted due to their significance while the rest with variations less than ±5% of the unity (normalized |Pr(λ)| at 550 nm) are not specified.

Fig. 7 Same as Fig. 6, but for the asteroids showing convex-shaped polarization spectra in Fig. 3.

4.2 Linking polarimetric observations to surface properties of asteroids

Multiple factors contribute to the optical properties of light scattered by asteroid regoliths. Research has long been conducted on how to enhance the NPB of atmosphereless small bodies. Laboratory experiments have shown that for low-albedo C-type analogs, a mixture of constituents in high albedo contrast develops a more pronounced depth and width of the NPB (Shkuratov 1987, 1992; Shkuratov et al. 1994). Shkuratov (1987) introduced the partitioning factor in his two-component mixture model as a means of assessing the albedo contrast between bright () and dark () materials. The author showed that a small-scale (the order of optical wavelengths) optical inhomogeneity (i.e., high k) of scatterers strongly amplifies the NPB in comparison to a single-component material. Recent laboratory studies (e.g., Spadaccia et al. 2022) validated the impact of this mixing effect on shaping the NPB. α₀ is also sensitive to the particle size of surface regoliths. The small (micron-to-submicron)-scale fine texture can affect the NPB parameters (Dollfus & Geake 1975; Dollfus et al. 1977) and, particularly, the NPB shows significant development in the presence of sub-micrometer grains on the surface (Geake & Geake 1990); otherwise, the NPB is generally invariant to dust aggregates with a size of up to centimeters (Gaffey 1974; Spadaccia et al. 2022 and references therein).

The apparent concordance between the relative intensity of the NPB (Fig. 5a) and the degree of aqueous alteration among different subclasses of C-complex asteroids (Fornasier et al. 2014; Kwon et al. 2022b) motivates us to interpret the continuous difference in polarimetric properties in line with the progressive alteration induced by hydration of asteroids (Rubin et al. 2007). In Appendix D, we compared laboratory measurements on anhydrous and hydrous regolith analogs and found supportive evidence that the hydrated samples tend to have larger α₀ than the anhydrous ones over a wide range of size distributions.

Aqueous alteration is a low-temperature process (Grimm & McSween 1989) in which melting ice interacts with native anhydrous silicates (olivine and pyroxene), forming phyllosilicates (mainly saponite or serpentine depending on heating duration and initial water-to-rock ratio; Zoensky et al. 1989; Kurokawa et al. 2020). Infrared spectroscopy leverages the center wavelength and shape of a 3-µm band to diagnose the hydration status of asteroids. A sharp increase in a continuum over 2.8–3.0 µm implying the band centered at <3 µm indicates significant amounts of phyllosilicates on the surface regoliths, whereas a round-shaped band centered at >3 µm indicate negligible and/or lack of aqueous alteration of the regoliths (Rivkin et al. 2022 and references therein). The former and the latter groups are referred to as “sharp type” (ST) and “non-sharp type” (NST), respectively, while the label for our C-complex asteroids is given in Table 2. For asteroids devoid of pronounced alteration evidence (such as B and F types), their groundmass composition has been supposed to be similar to anhydrous carbonaceous chondrites whose optical properties are predominated by opaque carbon grains (Gaffey 1974; Gaffey & McCord 1978; Rubin et al. 2007).

The shape of a reflectance spectrum provides insight into the optical dominance of spectrally opaque materials which may also be related to the degree of hydration of the surface regolith. McCord & Chapman (1975a,b) introduced the “bend parameter” as a way to measure the curvature of the short visible domains (i.e., bending near 500 nm). An asteroid having a lower bend parameter indicates higher opaques content (primarily carbon for C-complex asteroids). We measured the bend parameter of the large C-complex asteroids whose archival reflectance spectra are available (Fig. C.1). While the authors in the original studies used spectra ranging from 320 nm to 1100 nm, most asteroids in modern archives have not been observed below ~420 nm. We thus modified the equation to:

(5)

where R_X is the observed reflectance at the wavelength of X in nanometers. The second term in the first bracket on the right side was initially R₄₀₀ in McCord & Chapman (1975a,b) but replaced here with R₄₄₀ to mitigate the extrapolation uncertainty. As a result, the modification would reduce the contrast in curvature when compared to reality if an absorption feature exists in the UV region. The calculated bend parameters are tabulated in Table 2 and Fig. 8 shows their distribution as a function of α₀. The bend parameters are broadly distributed along the diagonal line from the lower left to the upper right. The average bend parameters are 0.03 ± 0.01, 0.02 ± 0.05, and 0.09 ± 0.02 for F-, B-, and Ch-type asteroids, respectively, whose increasing order is equivalent to the α₀ distribution in Fig. 5a. The background objects (asteroids in Fig. 5b) are also aligned with the global trend.

The observed subclass distribution in the α₀–bend parameter space would not be surprising if the surface composition is the primary factor. For F types having smaller α₀ values, the absence of a sharp decrease in reflectance (UV dropoff) down to 0.35 µm is due to its opaque-rich assemblage that makes the UV absorption of Fe²⁺ ions transition in silicates optically unrecognizable, yielding a small bend parameter (Burns 1970; Johnson & Fanale 1973; Gaffey & McCord 1978; Belskaya 2005). As regoliths undergo aqueous alteration (corresponding to a shift from F/B to Ch), the value of α₀ increases, while such opaque minerals decline in optical dominance. UV dropoff becomes more robust and its wing extends to visible wavelengths, thereby producing a large bend parameter. The above scenario can make α₀ and bend parameters are positively correlated with each other.

An alternative approach is to consider grain size effects. It is possible to qualitatively reproduce the observed bend parameter distribution of different subclasses in Fig. 8 if sizes of regolith constituents increase from Rayleigh-like sub-micrometer grains for F/B types (muting curvature) to ~a few ten micrometer-sized grains for Ch types. However, the bend and polarimetric parameters of the asteroids considered display no correlation with the albedo that is expected when grain size (and ensuing porosity) is the primary cause of diversity (Johnson & Fanale 1973; Emery & Brown 2004; Sultana et al. 2023), let alone any trends in the intensity of silicate emission features around 10 µm (Vernazza et al. 2012; Martin et al. 2022). Furthermore, we find that the size difference cannot explain the dichotomy in the correlation between polarization and reflectance spectra either (Fig. 6). Therefore, although it is beyond the scope of this paper to rigorously evaluate the impact of particle size on the observed trends, we prefer the scenario positing that the regolith composition, particularly the difference in optical heterogeneity of asteroid regoliths induced by aqueous alteration, would be the primary cause of the observed polarimetric and spectroscopic properties. For a complete description of scattering processes by asteroid surfaces, both composition and size aspects must be taken into account.

As a side note, the T-type asteroids in Fig. 8 have α₀ comparable to those of Ch-type asteroids, yet they have bend parameters that are as low as those of F/B-type asteroids. This would reflect the unique composition of the T-type asteroids (i.e., highly aqueously altered with abundant opaque minerals; Kwon et al. 2022b and references therein). It further suggests that α₀ of large C-complex asteroids may be a proxy for optical heterogeneity in proportion to the degree of aqueous alteration in the regolith, but not necessarily for the total amount of opaque materials present.

Fig. 8 Distribution of the modified bend parameter (from Gaffey 1974; McCord & Chapman 1975a,b) as a function of α0 of the C-complex asteroids in Fig. 5. The two parameters are provided in Table 2.

4.3 Potential of polarimetry as a tool for asteroid surface study

It is encouraging to observe that despite the narrow consideration of size and taxonomic classifications of asteroids in this study, their polarimetric dependence on phase angle and wavelength, as well as their relationship with reflectance spectra all demonstrate an appreciable degree of heterogeneity in their surface environments. We should emphasize that the polarimetric parameters of an asteroid’s regoliths are neither a simple nor a single-valued function for surface composition and the observational data are not yet sufficient to pinpoint a unique cause for the heterogeneity in the polarimetric behavior of large C-complex asteroids in the NPB. Nevertheless, the diversity of observations among the asteroids in this study suggests that the polarimetric properties of large C-complex asteroids in the NPB at optical wavelengths are primarily determined by their surface composition, which is strongly correlated with their hydration status.

Our tentative conclusion may also explain the α₀–diameter relationship seen in Fig. 5c. According to the existing α₀–particle size relationship (e.g. Geake & Dollfus 1986; Geake & Geake 1990), the smaller and larger α₀ values of F- and Ch-type asteroids are attributed to their bouldery (>millimeter) and fine-dust (≲sub-micrometer) regolith particles, respectively, which appears to contradict previous studies. However, if we look at size from the perspective of a dimension with a similar absorptivity (i.e., an imaginary part of the complex refractive index) on a wavelength scale, Ch-type regoliths can be viewed as clusters of small grains with high-albedo materials acting as a transparent matrix, whereas carbon-dominated F/B-type regoliths can be viewed as a single carbon grain. Light incident upon regoliths of the former and latter types will experience different numbers of interactions on a given scale, mimicking particles of small and large sizes, respectively. However, we note that this is only an educated guess at this point. A coordinated effort among laboratory and observational polarimetry studies is needed to understand better the mechanism behind light scattering on the asteroid surfaces.

Polarization parameters studied in our analysis seem to be applicable to smaller bodies, whose regolith size is expected to differ greatly from that of our target asteroids (Delbo et al. 2007; Marchis et al. 2012; Gundlach & Blum 2013). (101955) Bennu is a sub-kilometer-sized Near-Earth Asteroid (NEA) and the target of NASA’s sample return mission OSIRIS-REx (Lauretta et al. 2017). The NEA has small α₀ and |P_min| best-fit values (17°88 ± 0°40 and 1.43 ± 0.14%, respectively), corresponding to F-type asteroids (Cellino et al. 2018) and is apparently well located in the F-type regime in our Fig. 5a³.

Belskaya (2005) showed that 38-km F-type (302) Clarrissa has a small a0 that is within the range of most of the F-type asteroids. Another NEA, (3200) Phaethon, the target of JAXA’s DESTINY+ mission (Arai et al. 2018), is a ~5.1-km (Hanuš et al. 2016) B-type asteroid having α₀ of 20°2±0°2 and |P_min| of 1.2±0.1% (Devogèle et al. 2020), also marginally residing in the B-type regime in Fig. 5a.

Using α₀ value as a proxy to evaluate the relative primitive-ness of an asteroid’s surface appears consistent with a possible relationship between F-type asteroids and primitive surface environments (Kolokolova & Jockers 1997; Cellino et al. 2018). 133P/Elst-Pizarro is a ~4-km (Hsieh et al. 2009; Yu et al. 2020) main-belt comet, displaying comet-like dust ejection activity but dynamically in the asteroidal orbit (Hsieh & Jewitt 2006). Its NPB shape resembles that of F-type asteroids (Bagnulo et al. 2010) and likely the nucleus of 67P/Chyryumov-Gerasimenko (Stinson et al. 2016), casting the intriguing possibility of the association of F-type asteroids and a primitive comet nucleus whose surface is covered by an extremely low-tensile strength material (Attree et al. 2018) with water-ice chunks (Ciarniello et al. 2022; Fornasier et al. 2023). A recent mid-infrared study of F-type (704) Interamnia with the smallest α₀ in Fig. 5 estimates volume fraction of water ice of 9–66% (Jiang et al. 2023) supporting its primitive nature.

Lastly, combining the bend parameter and α₀ would help in the rough classification of F and B types. The absence of the UV dropoff information from the current archival data (SMASSII and S³OS²) makes F- and B-type asteroids hardly distinguishable from each other (Mahlke et al. 2022). Although the two subclasses are not sharply separated in Fig. 8, combining the two parameters is still able to isolate a moderate fraction of members of the two groups, thus encouraging further observations to increase the number of data sets available.

5 Summary

This paper reports the optical (400–950 nm) spectropolarimetric properties of 12 large C-complex asteroids. In conjunction with archival polarimetric and spectroscopic data, we investigate the polarimetric behaviors around the backscattering region and their correlation with the spectroscopic properties of a total of 64 asteroids. The main results are as follows:

1. The normalized polarization spectra |P_r(λ)| of 12 large C-complex asteroids observed with VLT/FORS2 and WHT/ISIS display color ranging from gray (0.01 ±0.01% per 100 nm) to blue (−0.12±0.01% per 100 nm) with a (quasi-)linear slope shortward of around 700 nm. The linearity extends to around 900 nm for most of the asteroids, except for three asteroids – T-type (308) Polyxo, Ch-type (404) Arsinoe, and C-type (444) Gyptis, which contain absorption bands in the so-called 0.7-µm and 3-µm region. The latter three have a concave |P_r(λ)| shape with a local polarization minimum between 600 and 750 nm, corresponding to their 0.7-µm band center position;

2. The overall V-band polarization-phase curves P_r(α) of our target asteroids in the NPB are broadly consistent, indicating the similarity of the bulk surface properties of the C-complex asteroids;

3. Despite the bulk similarity, there is an appreciable trend that different subclasses of the C-complex are distributed locally in the space of inversion angle and minimum polarization (α₀–|P_min|). The continuous alignment of F → B → T → Ch types in a direction of increasing both parameters ( and p = 0.0001) is particularly noteworthy since the ascending order reconciles the degree of aqueous alteration assessed by previous optical and infrared spectro-scopic studies. C/Xc-type asteroids fill the entire parameter space;

4. Asteroids in the particulate domain display weak, but meaningful inverse correlations with their diameters ( and p = 0.01), primarily driven by the distribution of C/Xc-type asteroids ( and p = 0.04). A larger sample of asteroids is needed to confirm this trend;

5. Asteroid albedos show no statistically significant relationship with any other parameters (including the NPB parameters, diameters, or orbital elements), which is in agreement with previous polarimetric studies;

6. For asteroids with sufficient variations in polarization, we observe a dichotomous relationship between their polarization and reflectance spectra. The spectra of absolute polarization degree |P_r(λ)| and a spectral slope S are positively correlated, but in an opposite direction: either having higher |P_r(λ)| and positive S or having lower |P_r(λ)| and negative S . The former group, represented by (308) Polyxo, has a concave polarization spectrum and relatively large α₀ values, whereas the latter group has a linear polarization spectrum and smaller α₀ values. None of the asteroids exhibit an inverse correlation between polarization and reflectance spectra like S-type asteroids (Bagnulo et al. 2015, 2023) that obey the Umov law;

7. The relationship of the bend parameter (modified in this study from the one originally defined by McCord & Chapman 1975a,b) and α₀ display a broad proportionality for the asteroids in our sample. Given that the bend parameter evaluates from the curvature around 500 nm the relative influence of opaque materials on reflectance spectra, this trend indicates the sequence of F → B → T → Ch-type asteroids in α₀ would reasonably well respond to their petrologic difference.

There is still a significant lack of information on the polarization distribution of asteroids covering a wide range of wavelengths, compared to other photometric or spectral archival data. For this reason, it is essential to put continuous efforts into building polarimetric catalogs (e.g., Bendjoya et al. 2022; Lupishko 2022; Masiero et al. 2022) in order to refine the interpretation of scattered light and eventually to define working observables for asteroid surfaces.

Acknowledgements

We thank an anonymous referee for providing valuable suggestions that led to substantial improvements. Discussions with Johannes Markkanen regarding light scattering in laboratory analogs were useful to elaborate the manuscript. Y.G.K gratefully acknowledges funding from the Volkswagen Foundation. Based in part on observations made with the ESO Very Large Telescope at the Paranal Observatory under program 095.C-0925 and 097.C-0853 (PI: Bagnulo) and collected at the WHT operated on the island of La Palma by the Isaac Newton Group during semester 14A.

Appendix A Selection of taxonomic types of asteroids in Table 2

The purpose of this section is to demonstrate our criteria for selecting the taxonomic types of asteroids. We refer to the taxonomic types of the two classification systems in the first place from the table provided by Small-Body Database Lookup⁴ and then apply the following criteria to re-classify asteroids that have been designated as different subclasses by Tholen (Tholen 1984) and Bus-DeMeo (SMASSII; DeMeo et al. 2009) classification systems.

Firstly, since the contemporary SMASSII catalog does not cover the UV-short visible region, in which Tholen F types exhibit a different spectral slope from SMASSII B types, Tholen F types were grouped with SMASSII B types. We thus always prioritized Tholen F types over SMASSII B types unless both classification systems assign B to an asteroid (e.g., (372) Palma).

Secondly, SMASSII X types contain a higher heterogeneity of member asteroids, as the taxonomic scheme uses only spectral shapes and does not incorporate albedo information. Tholen P types fall in this spectral group. SMASSII X types’ spectral slopes distribute over a wide range from a neutral-to-slight-red (C-type) slope to a moderately red (Tholen P-type) slope (DeMeo et al. 2009). Therefore, for asteroids assigned in Tholen P types and/or SMASSII X, Xc, or Xk types, we re-classified them based on the mean reflectance data available on the SMASSII webpage⁵ and previous X-type spectroscopic observations by Fornasier et al. (2011). If a spectral slope of an asteroid over 500800 nm is ≳2.5% per 100 nm, we assigned it P and otherwise Xc. Xc types were illustrated in Figures 5 and 8 using the same symbol as the C types, given that their spectral characteristics are similar to those of the C-type asteroids (DeMeo et al. 2009, 2015). C types contain features shared by several subclasses and were therefore ranked lowest.

Lastly, the technical definition of SMASSII Ch types (and some of Tholen G types) is when an asteroid has an absorption band in the so-called 0.7-µm region associated with phyllosilicates. However, this feature has been observed not only in Ch-type asteroids, but also in other taxonomic subclasses, and sometimes a single asteroid shows variations in its band depth at different epochs of observations (e.g., Fornasier et al. 2016). Given that this wavelength domain overlaps with the atmospheric H2O absorption band (Lord 1992), thereby subject to systematic uncertainties introduced by calibration of observations, the 0.7-µm band feature reported for some asteroids would be artifacts (e.g., Vilas & Gaffey 1989). In this study, if an asteroid has ever been diagnosed as Ch by either of the two taxonomies, we assigned it Ch. However, for asteroids that have a 0.7-µm absorption feature but belong to non-Ch types, we determined their taxonomies considering infrared spectroscopic observations together: if such an asteroid has a V-shaped absorption feature whose band center is <3 µm (phyllosilicate-dominated “ST” in Table 2), we took its 0.7-µm band as valid evidence for the hydrated status of the asteroid and thus designated it as a Ch type. On the contrary, if an asteroid has a 0.7-µm feature but not an ST-like infrared absorption (either having an infrared band center at >3 µm, that is, “NST” in Table 2, or no available infrared data), we kept its non-Ch spectral type.

Asteroids whose taxonomic type was updated on the basis of the above criteria have starred footnotes (★) in Table 2. Below is special remarks for three asteroids.

• (54) Alexandra and (386) Siegena: Both are Tholen C and SMASSII C types. Despite their non-Ch type designations, their reflectance spectra contain both a discernible 0.7-µm absorption feature (Fornasier et al. 2014) and an ST-like phyllosilicate absorption band (Takir & Emery 2012; Rivkin et al. 2022). We thus assigned them to be Ch.

• (62) Erato: This is Tholen BU and SMASSII Ch type. The asteroid is a member of the Themis asteroid family (Nesvorný et al. 2015) and should be considered a Ch type since SMASSII taxonomy already designates the asteroid as Ch. However, given the absence of a 0.7-µm absorption feature and its overall affinity to the B-type reflectance spectra (Yang 2009 and references therein), we considered this asteroid as a B subclass.

Appendix B Statistical estimation to test for the correlations in Figure 5

To discern whether a set of two variables in Figures 5a and 5c are correlated, we conducted Spearman’s rank correlation test (Spearman 1904) whose correlation coefficient ρ and p-value p were estimated using Monte Carlo simulation. Given a data set consisting of N data pairs, X ± σ_X and Y ± σ_Y, each x_i and y_i pair was randomly generated in the Gaussian distribution whose standard deviation equals to the 1-σ uncertainty, σ_X and σ_Y, of the central values. For statistical significance, we tested i = 10,000 Monte Carlo particles creating the probability distribution of ρ and p.

Figure B.1 shows the distribution of 10,000 clones generated to test the correlation of α₀−|P_min| in Figure 5a. In a 68.2% confidence interval, the median of the distribution is and its mode is 0.72. The resulting p-value is p = 0.0001, confirming the proposed correlation’s statistical significance (>3σ). Likewise, the identical scheme of the Monte Carlo method reveals that and its mode of 0.25 for F types, and its mode of 0.42 for B types, and and its mode of 0.33 for Ch types. T types were not analyzed due to the small data number (N = 2). Similar to the trend of F–B–T–Ch–type asteroids as a whole, members of asteroids in each taxonomic type are in a positive correlation between the NPB parameters, but with less statistical significance.

The correlation test between the inversion angle α₀ and the diameter of asteroids in Figure 5c is also shown in Figure B.2. For the asteroids whose α₀ values lead them to belong in the particulate domain (Geake & Dollfus 1986), the median of the distribution as a whole in a 68.2% confidence interval is and its mode is −0.46, with a p-value of 0.01. This trend seems largely driven by the distribution of C-type asteroids ( and its mode of -0.52, with a p-value of 0.04) distributed in the widest range of diameters.

Fig. B.1 Number distribution of 10,000 clones randomly generated within the range corresponding to the errorbars of the inversion angle α0 and the absolute value of the minimum polarization degree |Pmin|. The retrieved Spearman’s rank correlation coefficient ρ histogram is present in panel b. Colors do not use the same color code used in Figure 5a and are instead assigned at random in the chosen color map.

Fig. B.2 Same as Figure B.1 but the test for the correlation of α0– Diameter in Figure 5c.

Appendix C Archival reflectance spectra of the asteroids in Table 2

This section summarizes the reflectance spectra normalized at 550 nm used in the analysis in the main text. In Figure C.1, the spectra are given in ascending order of the number of asteroids.

Fig. C.1 Reflectance spectra of large, low-albedo C-complex asteroids in Table 2 normalized at 550 nm from SMASSII (Bus & Binzel 2002; DeMeo et al. 2009) and S3OS2 (Lazzaro et al. 2004). Numbers in the parenthesis in each panel indicate phase angles at the time of observations. Weak features near 900 nm appear in the spectra taken at different times with different instruments at edge wavelengths where detectors are less sensitive. Discontinuities shown in several SMASSII spectra near 800 nm are caused by the concatenation of different datasets (DeMeo et al. 2009).

Appendix D Comparison of the laboratory experiments on the NPB properties of anhydrous and hydrous regolith analogs

Laboratory measurements of samples with properties that are similar to those before and after aqueous alteration may provide additional information regarding the α₀ development. As such, we utilized experiment data from PROGRA² (the French acronym for Optical Properties of Astronomical and Atmospheric Grains⁶). The database provides polarization and brightness phase curves for irregular aggregates suspended or deposited on surfaces over 450–1600 nm and has been widely used to study light scattered by small Solar System objects and Martian regoliths (Worms et al. 2000; Hadamcik et al. 2006; Levasseur-Regourd et al. 2015; Hadamcik et al. 2023).

In Figure D.1a, we show the polarization-phase curves of two martian soil simulants (consisting of mainly silica and Fe oxides, with a size distribution of <100 µm and 250–1000 µm, each), Fe-silica with carbon (with a size distribution, peaked at ~50 µm having a long tail up to ~300 µm), and Mg·Fe-silica (25% of each) with carbon (14 nm, 50% in abundance) with a size distribution peaked at ≲20 µm. The last two samples are analogs of F-type asteroids (Gaffey & McCord 1978; Allen et al. 1998; Hadamcik et al. 2003) in that their optical properties are dominated by intimately mixed opaques (nanometer-sized carbon) and thus exhibit decent bulk optical homogeneity. Their polarization-phase curves are highly similar despite the Mgsilica in the latter samples, which makes sense given that carbon is so optically dominant that even small amounts (a few percent) are sufficient to flatten and mute the spectrum and lower the albedo well under 0.1 (Johnson & Fanale 1973; Gaffey 1974; Mukai 1986). The Martian simulant (composed of olivines, pyroxene, and oxides) contains OH bands on its VNIR reflectance spectrum, albeit not exactly produced by phyllosilicates (Allen et al. 1998; Hadamcik et al. 2003). We thus expect the Martian simulant to exhibit greater optical heterogeneity at the microscale than the silica samples, qualitatively mimicking a composition post-aqueous alteration (higher optical heterogeneity of the constituents with hydrous bands), especially when compared to the other two F-type analogs.

Despite having significantly different size distributions, samples in comparable optical heterogeneity possess nearly identical α₀ values (Fig. D.1). α₀ values are consistently smaller for optically homogeneous, small-sized opaque-governing samples (grey lines) than for optically heterogeneous, oxide-containing samples (black lines), in agreement with recent laboratory experiments (e.g., Spadaccia et al. 2022). The polarization color in all samples in the NPB is blue over 545–630 nm (panel b in Fig. D.1), which is consistent with the observations of our target asteroids (Fig. 3). The samples compared here may not be suitable for testing the impact of submicron-sized particles on the NPB shape due to the practical reasons for sample preparation (J. Markkanen, personal communication); nonetheless, the observed α₀ differences between samples with varying degrees of carbon impact on the wavelength-scale support the hypothesis that a0 would tie to the compositional properties of C-complex asteroids.

Fig. D.1 Polarization-phase curves of martian simulants and mixtures of Mg-Fe silica with carbon from the PROGRA2 experiment database. Sample names in the legend are the same as those in the database. Data in panel a are measurements at 545 nm. Filled symbols in panel b are the same as those in panel a (i.e., the Martian simulant (<100 µm) and Mg·Fe-silica with carbon at 545 nm) but their open counterparts indicate measurements at 630 nm.

References

All Tables

All Figures

	Fig. 1 Orbital distributions of 12 C-complex asteroids observed with VLT/FORS2 in spectropolarimetry Background grey dots are numbered main-belt asteroids that are quoted from JPL Small-Body Database Query. Symbols in the highlighted color are proportional to the size of asteroids. Vertical lines at different heliocentric distances indicate the most important mean-motion resonances with Jupiter.
In the text

Fig. 2 Reduced polarization degree, Pr, and its position vector, θr, as a function of wavelength for 12 C-complex asteroids taken at 17 epochs (Table 1). In each panel, the phase angle (α) at the time of observation is provided in the legend. Different colors were used for asteroids observed more than once. The curve with salmon-colored diamonds represents the binned polarization spectrum with the coarsest spectral resolution (an interval of ~20–40 nm).

In the text

	Fig. 3 Best-fitting quadratic curves of polarization spectra of the target asteroids normalized at 550 nm. Three asteroids showing concave shapes – (308) Polyxo, (404) Arsinoe, and (444) Gyptis - are visualized in a separate window.
In the text

Fig. 4 V-band polarization-phase curve Pr(α) of our 12 target asteroids. Symbols with thick line edges represent the data measured from this study, with gray-scale colors indicating their relative albedos ranging from white (high-albedo end: 0.094 of (175) Andromache) and black (low-albedo end: 0.037 of (404) Arsinoe). The albedos were from the NASA JPL Small-Body Database. Background circles are their archival data from the NASA Planetary Data System (PDS) Small Bodies Node (Lupishko 2022) and Calern Asteroid Polarization Survey (CAPS; Bendjoya et al. 2022). Points of the dwarf planet (1) Ceres are distinguished by black-edged orange circles.

In the text

Fig. 5 Distribution of 64 large C-complex asteroids in the retrieved inversion angle α0 and absolute polarization minimum |Pmin| space (panels a and b), along with the size of the asteroids (panel c). Guidelines in panels a and b represent the borders of surface regimes dominated by different-size regolith particles (Geake & Dollfus 1986): the left border separates the boulder and bare-rock regoliths (leftmost) and particular surface domain (intermediate) and the right one divides the particulate domain and fine-dust regoliths (rightmost). Different colors are assigned to different taxonomic types. Panel c illustrates the distribution of asteroids in the particulate domain.

In the text

Fig. 6 Correlation between the spectra of the normalized absolute polarization degree |Pr(λ)| and a spectral slope S of reflectance spectra for the C-complex asteroids in Fig. 3. Both parameters were normalized at 550 nm and measured in each spectral bin. Circles and diamonds of each curve indicate the values derived in the shortest (450–550 nm) and longest (750–900 nm) bins, respectively. Including those endpoints, a single line connecting a total of four points represent |Pr| and S values of an asteroid evaluated over 450–550 nm, 550–650 nm, 650–750 nm, and 750–900 nm. Only three asteroids – (10) Hygiea, (308) Polyxo, and (451) Patientia – are highlighted due to their significance while the rest with variations less than ±5% of the unity (normalized |Pr(λ)| at 550 nm) are not specified.

In the text

	Fig. 7 Same as Fig. 6, but for the asteroids showing convex-shaped polarization spectra in Fig. 3.
In the text

	Fig. 8 Distribution of the modified bend parameter (from Gaffey 1974; McCord & Chapman 1975a,b) as a function of α0 of the C-complex asteroids in Fig. 5. The two parameters are provided in Table 2.
In the text

Fig. B.1 Number distribution of 10,000 clones randomly generated within the range corresponding to the errorbars of the inversion angle α0 and the absolute value of the minimum polarization degree |Pmin|. The retrieved Spearman’s rank correlation coefficient ρ histogram is present in panel b. Colors do not use the same color code used in Figure 5a and are instead assigned at random in the chosen color map.

In the text

	Fig. B.2 Same as Figure B.1 but the test for the correlation of α0– Diameter in Figure 5c.
In the text

Fig. C.1 Reflectance spectra of large, low-albedo C-complex asteroids in Table 2 normalized at 550 nm from SMASSII (Bus & Binzel 2002; DeMeo et al. 2009) and S3OS2 (Lazzaro et al. 2004). Numbers in the parenthesis in each panel indicate phase angles at the time of observations. Weak features near 900 nm appear in the spectra taken at different times with different instruments at edge wavelengths where detectors are less sensitive. Discontinuities shown in several SMASSII spectra near 800 nm are caused by the concatenation of different datasets (DeMeo et al. 2009).

In the text

Fig. D.1 Polarization-phase curves of martian simulants and mixtures of Mg-Fe silica with carbon from the PROGRA2 experiment database. Sample names in the legend are the same as those in the database. Data in panel a are measurements at 545 nm. Filled symbols in panel b are the same as those in panel a (i.e., the Martian simulant (<100 µm) and Mg·Fe-silica with carbon at 545 nm) but their open counterparts indicate measurements at 630 nm.

In the text