11institutetext: Université Côte d’Azur, CNRS–Lagrange, Observatoire de la
Côte d’Azur, CS 34229 – F 06304 NICE Cedex 4, France
11email<EMAIL_ADDRESS>22institutetext: Southwest Research Institute, 1050
Walnut St. Suite 300, Boulder, CO, 80302, USA

# Gaia view of primitive inner-belt asteroid families

Searching for the origins of asteroids Bennu and Ryugu

M. Delbo 11 C. Avdellidou 11 K. J. Walsh 22

(Received September 15, 1996; accepted March 16, 1997)

###### Abstract

Aims. Near-Earth asteroids Ryugu and Bennu, were visited, characterised, and
sampled by the Hayabusa2 and OSIRIS-REx missions, where remote sensing data
and sample return analysis showed that both asteroids have primitive, hydrated
and organic-rich compositions. The dark families of the inner main belt that
belong to the spectroscopic C-complex have been claimed to be the sources of
both Ryugu and Bennu, hence there have been large efforts to spectroscopically
characterise them by ground-based observations.

Methods. Here we used the Gaia Data Release 3 (Gaia DR3) asteroid reflectance
spectra in order to characterise the 11 known inner main belt C-complex
families (Chaldaea, Chimaera, Clarissa, Erigone, Eulalia, Klio, Polana,
Primordial, Sulamitis, Svea, Tamara), using space-borne visible-light
spectroscopic observations. For each family we extracted the family members
that have known geometric visible albedo values and Gaia DR3 data, and we
created an average reflectance spectrum per family, between 370 and 950
$\text{\,}\mathrm{nm}$. These averages were then compared with the ground-
based visible spectroscopic surveys of the same families, and to Bennu’s and
Ryugu’s space and ground-based spectra in the same wavelength range.

Results. Gaia DR3 reflectance spectra of the dark asteroid families of the
inner main belt are in general consistent with previous findings. The only
exception is the case of the Svea family: previous surveys classified its
members as B-types, whereas the average reflectance spectrum from Gaia DR3 is
similar to a C-type. We also showed that the Polana and the Eulalia families
can be distinguished in the wavelength region 370 – 500
$\text{\,}\mathrm{nm}$. Among all the primitive inner main belt families, we
found that the average reflectance spectra of the Eulalia and Polana families
are the most similar to those of Bennu and Ryugu, respectively. In particular,
Eulalia family’s average spectrum is a good match to Bennu’s in the wavelength
range 450 – 800 $\text{\,}\mathrm{nm}$, while beyond 800
$\text{\,}\mathrm{nm}$ the spectrum of Bennu is bluer than that of Eulalia.
Moreover, the spectrum of the Polana family has the smallest discrepancy
(smallest $\chi^{2}$) against the spectrum of Ryugu, although this match is
formally unsatisfactory (reduced $\chi^{2}\sim$ 1.9).

###### Key Words.:

minor planets, asteroids:general – astronomical databases:miscellaneous –
techniques: spectroscopic

##  1 Introduction

The main belt consists of over one million known asteroids. These have a
plethora of compositions and values of albedo, ranging from few to several
tens of percent. The two largest compositional classes are those broadly
identified by the so-called spectroscopic S- and C-complexes (DeMeo et al.,
2015). These spectral complexes are also separated in albedo, with the S- and
C-complex asteroids having, in general, geometric visible albedo ($p_{V}$)
values smaller than or larger than 0.12, respectively (Delbo et al., 2017;
Ferrone et al., 2023).

Recent theories of asteroid formation invoke an in situ (or quasi-in situ)
origin of the S-complex bodies (Walsh et al., 2011; Raymond & Izidoro, 2017b).
On the other hand, C-complex asteroids are supposed to be exogenous to the
main belt. It is generally thought that these bodies accreted from the dust of
the protoplanetary disk in the giant planet region (Warren, 2011; Kruijer et
al., 2017; Nakamura et al., 2023), were later transported, and finally
implanted into the main belt from larger heliocentric distances. A handful of
mechanisms have been proposed for this process, including the rapid growth of
the giant planets scattering nearby planetesimals (Raymond & Izidoro, 2017a),
the migration of the giant planets amidst the solar nebula (Walsh et al.,
2011; Pirani et al., 2019) and likely some contributions from the giant planet
instability (Levison et al., 2009; Vokrouhlický et al., 2016).

After the original asteroids (also known as planetesimals; Delbo et al., 2017)
were accreted (for the S-complex) or implanted (for the C-complex) in the main
belt, collisions with other asteroids throughout the history of the solar
system have broken several of them, which in turn created families of asteroid
fragments (Nesvorny et al., 2015). Hence, the large majority of asteroids are
collisional fragments of the original ones (Delbo et al., 2017; Dermott et
al., 2018; Ferrone et al., 2023).

This large population of small fragments can reach the near-Earth space via
well-established dynamical routes and eventually arrive to Earth as meteorites
(Greenwood et al., 2020, and references therein). The carbonaceous chondrite
meteorites (CCs) are typically associated with the C-complex asteroids. The
study of CCs has shown that the C-complex asteroids carry a primitive
composition, enriched in water and organics, and were possibly responsible for
delivering the ingredients of life to primeval Earth via impacts (Chyba &
Sagan, 1992). On the other hand, despite the fact that the C-complex asteroids
are a majority in the main belt, the CCs are a small fraction in our
meteoritic inventory (Norton & Chitwood, 2008). This is reasonable if we
consider the filtering by Earth’s atmosphere. Due to the fact that C-complex
asteroid materials are mechanically quite weak and some of them with large
porosity (Grott et al., 2019; Ballouz et al., 2020; Cambioni et al., 2021), it
is very difficult to survive the atmospheric passage at the typical velocities
that meteoroids hit our planet.

The above have also been confirmed by Hayabusa2 (JAXA) and OSIRIS-REx (NASA)
sample return missions, which have visited, characterised, and sampled the
near-Earth C-complex asteroids (162173) Ryugu (Kitazato et al., 2019) and
(101955) Bennu (Lauretta et al., 2019), respectively. Recent sample analysis
has shown that Ryugu is an aqueous altered asteroid, with a CI (Ivuna-like)
(Yokoyama et al., 2023), organic-rich composition (Ito et al., 2022; Naraoka
et al., 2023; Yabuta et al., 2023). Moreover, Ryugu samples provide an
excellent example of primitive solar system material that is ”clean” from
terrestrial contamination. Although Bennu’s sample has not arrived on Earth
yet, OSIRIS-REx spectroscopic data from the visible to thermal infrared
wavelengths have shown that the surface is consistent with an aqueously
altered CM chondrite (Hamilton et al., 2019; Kaplan et al., 2020; Simon et
al., 2020).

It is established that, based on the orbits of Ryugu and Bennu, both asteroids
escaped the inner-main belt (IMB) via the $\nu_{6}$ secular resonance (Campins
et al., 2010, 2013; Bottke et al., 2015) and reached the near-Earth space
about 5 (Okazaki et al., 2023) and 1.75 Myr ago (Ballouz et al., 2020),
respectively. Therefore, their source regions (i.e., C-complex asteroid
families), should be located in the inner main belt and could possibly have
different compositions.

In the inner main belt 11 dark C-complex asteroid families of different sizes
and ages are known to exist (Walsh et al., 2013; Nesvorny et al., 2015; Delbo
et al., 2017), namely: Polana, Eulalia, Erigone, Chaldaea, Chimaera, Clarissa,
Klio, Sulamitis, Svea, Tamara, and the so-called Primordial one (the age of a
family is given by the epoch of the collision that created it; Nesvorny et
al., 2015). Since 2016, the majority of these families have been the subject
of dedicated ground-based visible and near-infrared spectroscopic observations
to characterise their composition, report on the level of their homogeneity,
and compare to each other. This effort has been mostly driven by the need to
constrain the source regions of Ryugu and Bennu, to give astronomical context
for the returned samples (e.g., Lauretta et al., 2015).

Specifically, it has been reported that any slope difference between the
Polana and Eulalia family members in the visible light range (VIS) is within
1$\sigma$ uncertainty (de León et al., 2016), while they are neither
distinguishable in the near-infrared range (NIR) (Pinilla-Alonso et al.,
2016), nor in the near-ultraviolet (NUV) (Tatsumi et al., 2022). Visible
spectroscopic observations of Erigone, Clarissa, Sulamitis, Klio, Chaldaea,
and Svea (Morate et al., 2018, 2019), showed that the slopes of Clarissa and
Svea family members agree with the Polana family members, and, similarly,
Sulamitis family members’ slopes to Erigone ones. Despite their difference in
VIS slopes, NIR observations showed that Sulamitis and Klio members are close
to Polanas (Arredondo et al., 2020, 2021b). In addition, Klio and Chaldaea
show a complementarity in VIS slopes that together could match the Erigone’s
spectral slope distribution, while in the NIR appear to have extremely similar
slopes. It is proposed that these findings could indicate a common origin to
these families (Arredondo et al., 2021a), an idea that can be supported by the
fact that both families overlap in the space of proper orbital semi-major axis
vs. inverse diameter ($a$ vs. 1/$D$). Another interesting result is that the
dark IMB families have been divided into two broad groups, the so-called
”blue” families with blue to moderate slopes and no sign of hydration, and the
”red” families, which include objects with the 700 $\text{\,}\mathrm{nm}$
absorption band present (Morate et al., 2019).

In this work, our goal is to study the IMB dark primitive families by
exploiting hundreds of VIS spectra from Gaia Data Release 3 (DR3) catalogue
(Gaia Collaboration et al., 2023), in order to understand their differences
and similarities. In this way we will obtain a view of their original
asteroids (planetesimals). In section 2 we will present the datasets that
include the family membership and the Gaia DR3 spectra of these populations;
in sections 3 our analysis and results, while in section 4 the discussion on
our findings, including a comparison with the ground-based spectroscopic
studies.

##  2 Data

###  2.1 Gaia DR3 spectroscopic data

We made use of the reflectance spectra that were derived from asteroid
spectroscopic observations obtained by Gaia between 5 August 2014 and 28 May
2017 and were published in June 2022 as part of the DR3 (Gaia Collaboration et
al., 2023). This dataset consists of mean reflectance spectra in the VIS
wavelength range of 60,518 Solar System objects (SSOs). The reflectance
spectra were acquired by two low-resolution slit-less spectrophotometers on
board Gaia, the blue and red spectrophotometers (BP and RP), which are
respectively optimised for the blue and red part of the spectrum.
Specifically, the BP spans the wavelength range from 330 to 680
$\text{\,}\mathrm{nm}$ and the RP covers the range from 640 to 1050
$\text{\,}\mathrm{nm}$. The spectral resolution of each spectrophotometer is a
function of wavelength, and varies from 4 to 32 $\text{\,}\mathrm{nm}$ pixel-1
for the BP and 7 to 15 $\text{\,}\mathrm{nm}$ pixel-1 for the RP (Gaia
Collaboration et al., 2023; Carrasco et al., 2021; Jordi et al., 2010). When
an asteroid transited on the focal plane of Gaia at a given epoch, each
spectrophotometer measured photons at every wavelength to create ‘epoch
spectra’. For each asteroid, each epoch spectrum was divided by the mean
spectrum of a series of trusted solar analogue stars (see Table C.1 of Gaia
Collaboration et al., 2023) in order to create ”epoch reflectances”. Given
that the wavelength range of both instruments overlaps in the 650-680
$\text{\,}\mathrm{nm}$ interval, the two epoch reflectances were merged to
create a full epoch reflectance. To each asteroid was finally associated a
unique mean reflectance spectrum obtained by averaging several epoch
reflectances, spanning the visible wavelength range from 374 to 1034
$\text{\,}\mathrm{nm}$ in 16 discrete wavelength bands (Gaia Collaboration et
al., 2023). A ‘reflectance_spectrum_flag’ (hereafter RSF) number was also
associated with each band, assessing the estimated quality of the band. In
some cases, the merging of the epoch spectra taken by each spectrophotometer
was not perfect, which can lead to the creation of artefact bands (Gaia
Collaboration et al., 2023). Thus, caution must be taken when analysing the
mean reflectance spectra in the overlapping wavelength interval. In a similar
way, the bluest and reddest data bands of Gaia spectra could also be affected
by systematics due to the low efficiency of the spectrophotometers in these
bands. They were not always flagged but they need to be taken with caution as
well (Gaia Collaboration et al., 2023; Galinier et al., 2023).

###  2.2 C-complex inner main belt families

We retrieved the asteroid membership for seven out of 11 IMB C-complex
families from Nesvorny et al. (2015) (see Table 1). The latter family
identification within the main belt asteroid population is, by construction,
conservative, meaning that good separation between the families is ensured,
limiting therefore the number of family interlopers. This is very good for our
purposes of studying similarities and differences between families in the same
region of the main belt. For the Polana and Eulalia families we used the
membership as it was defined in Walsh et al. (2013). For the Tamara family,
which is located in the high-inclination Phocaea region, we used the
membership of Novaković et al. (2017). Finally, for the so-called Primordial
family we used the membership defined by Delbo et al. (2017). We retrieved the
$p_{V}$-values, where applicable, from the Minor Planet Physical Properties
Catalogue (MP3C)111https://mp3c.oca.eu. Specifically, for each asteroid the
$p_{V}$-values reported by the MP3C are averaged values of the published
geometric visible albedo determination (a weighting factor equal to the
inverse of square of the published formal uncertainty of each albedo
determination is used by the MP3C for their averaging).

##  3 Analysis & Results

###  3.1 Removal of family interlopers

It is important to remember that the identification of family members based on
clustering in orbital elements (Nesvorny et al., 2015) or correlations between
the orbital proper semi-major axes and the $1/D$ (V-shape criterion Bolin et
al., 2017; Delbo et al., 2017) is a rough expression of the true membership
(Nesvorny et al., 2015; Ferrone et al., 2023): in fact, asteroids unrelated to
a family that happen to have values of proper elements and $1/D$-values within
the orbital and/or $1/D$ range of that family are grouped together with the
true members. These objects identified as ”false positive” are typically
called family interlopers (Nesvorny et al., 2015). Interlopers can be
distinguished amongst family members by their anomalous spectral properties or
albedo values compared to the majority of the family members.

To filter out potential family interlopers and ensure the cleanest sample for
our spectral analysis of each family, we applied a series of criteria on the
values of the $p_{V}$, spectral slope ($s$), and reflectance difference
$R_{z}-R_{i}$ of family members. First of all, we required the family members
to have $p_{V}$-values reported in the literature, and be $p_{V}$ ¡ 0.12. In
addition, we required the Gaia DR3 spectra to have -5 ¡ $s$ ¡ 6 %/100
$\text{\,}\mathrm{nm}$, and -0.2 ¡ $R_{z}-R_{i}$ ¡ 0.185. The selected $p_{V}$
threshold has been shown to separate the dark from the bright asteroids
respectively belonging to the C- and S-complexes (Delbo et al., 2017). In
particular, 88% of the C-complex asteroids are contained within the asteroid
population with $p_{V}$¡0.12 (Delbo et al., 2017). Moreover, the ranges of
spectral parameters mentioned above define the boundaries of the C- and
B-classes and therefore of the spectroscopic C-complex (DeMeo & Carry, 2013).
The application of these criteria results in reducing the number of members of
each family included in our analysis; the latter number is reported, for each
family, in Table 1.

The spectral parameters were calculated following the method of Gaia
Collaboration et al. (2023), in two steps. First, the spectral slope was
determined as the angular coefficient of the best-fit straight line to the
reflectance data between 450 and 600 $\text{\,}\mathrm{nm}$. Then, the
reflectance difference $R_{z}-R_{i}$ was obtained by fitting a natural
smoothing spline, $S(\lambda)$, (Python3 module CSAPS; smoothing coefficient
of $5\times 10^{-7}$) and then by assuming that
$R_{z}-R_{I}=S(\lambda_{z})-S(\lambda_{i})$, where $\lambda_{z}$ = 893.2
$\text{\,}\mathrm{nm}$ and $\lambda_{i}$ =748.0 $\text{\,}\mathrm{nm}$.

Table 1: Input dataset.

Family | $N_{members}$ | $N_{DR3}$ | $N_{filtered}$ | $p_{V}$ |  $\sigma$${}_{p_{V}}$ | $s$ |  $\sigma$s | $R_{z}-R_{i}$ |  $\sigma$${}_{R_{z}-R_{i}}$ | Ref.
---|---|---|---|---|---|---|---|---|---|---
Chaldaea | 132 | 40 | 32 | 0.067 | 0.019 | +1.30 | 2.53 | 0.066 | 0.035 | Nesvorny et al. (2015)
Chimaera | 108 | 18 | 11 | 0.054 | 0.014 | +2.65 | 1.39 | 0.099 | 0.041 | Nesvorny et al. (2015)
Clarissa | 179 | 2 | 2 | 0.069 | 0.002 | -1.00 | - | 0.036 | - | Nesvorny et al. (2015)
Erigone | 1776 | 142 | 82 | 0.056 | 0.017 | +1.75 | 2.14 | 0.053 | 0.035 | Nesvorny et al. (2015)
Eulalia | 1624 | 248 | 205 | 0.057 | 0.012 | +1.02 | 2.03 | 0.038 | 0.042 | Walsh et al. (2013)
Klio | 330 | 88 | 72 | 0.067 | 0.016 | +2.23 | 2.17 | 0.061 | 0.039 | Nesvorny et al. (2015)
Polana | 2037 | 577 | 243 | 0.058 | 0.016 | +1.01 | 2.31 | 0.035 | 0.045 | Walsh et al. (2013)
Primordial | 118 | 89 | 64 | 0.061 | 0.022 | +1.93 | 2.29 | 0.043 | 0.048 | Delbo et al. (2017)
Sulamitis | 303 | 32 | 23 | 0.056 | 0.010 | +3.14 | 1.99 | 0.071 | 0.038 | Nesvorny et al. (2015)
Svea | 48 | 4 | 2 | 0.060 | 0.019 | +1.42 | - | 0.006 | - | Nesvorny et al. (2015)
Tamara | 226 | 56 | 56 | 0.060 | 0.014 | +1.60 | 2.11 | 0.060 | 0.042 | Novaković et al. (2017)

222The symbols $N_{members}$, $N_{DR3}$, and $N_{filtered}$ represent,
respectively, the number of family members, the number of family members with
Gaia DR3 reflectance spectra and the number of family members used in our
analysis after the application of the filters described in the main text. The
symbols $p_{V}$, $\sigma$${}_{p_{V}}$, $s$, $\sigma$s, $R_{z}-R_{i}$, and
$\sigma$${}_{R_{z}-R_{i}}$ represent, respectively, the average geometric
visible albedo, the standard deviation of the geometric visible albedo, the
reflectance spectral slope between 450 and 750 $\text{\,}\mathrm{nm}$, the
standard deviation of the slope, the difference between the reflectance at
893.2 and at 748.0 $\text{\,}\mathrm{nm}$, and the standard deviation of the
difference of reflectance for the distribution of the members of the different
families filtered as explained in the text. The column Ref. reports the
reference from which we obtained the initial family membership definition.

###  3.2 Spectra profiles of IMB dark families

Once the family members were filtered for potential interlopers, we calculated
the average reflectance spectrum of each family. First, for each reflectance
spectrum, we removed those data points with a ‘reflectance_spectrum_flag’
(RSF) value $>$ 0, which indicates that those values are suspected to be of
poorer quality (Gaia Collaboration et al., 2023). In addition, we did not take
into account all spectra data at 990 and 1034 $\text{\,}\mathrm{nm}$ because
the reflectance at these wavelengths is typically affected by a systematic
increase (Gaia Collaboration et al., 2023). Next, we calculated the weighted
average at each of the 16 wavelength-bands with which the Gaia DR3 reflectance
spectra were expressed (Gaia Collaboration et al., 2023). We also calculated
the median average deviation at each wavelength, using $1/\sigma^{2}$ as
weights, where $\sigma$ is the reflectance uncertainty reported in Gaia DR3.
After having rejected those data whose distance from the mean is larger than
2.5$\times$ the median average deviation, we recalculated the weighted mean
and its uncertainty. The uncertainty of the average reflectance spectra were
calculated using a bootstrap technique: namely, we iteratively randomly
selected 75% of the filtered family and recalculated an average reflectance
spectrum and, after having reached 1000 iterations, we calculated the standard
deviation of the mean spectra at each wavelength. Finally, we applied the
correction in the blue region of the spectrum, following the procedure of
Tinaut-Ruano et al. (2023), where we multiplied by 1.07, 1.05, 1.02, and 1.01
the reflectance values at the wavelengths of 374, 418, 462, and 506
$\text{\,}\mathrm{nm}$, respectively. The resulting average reflectance
spectra and their uncertainties for each family are shown in Fig. 1.

Figure 1: Average Gaia DR3 spectra of the 11 inner main belt dark families
that belong to the spectroscopic C-complex. Spectra are shifted on the
reflectance axis for better visibility. The error bars are included, but are
smaller than the marker size in most cases.

##  4 Discussion

There appear to be two classes of dark families in the inner main belt.
Specifically, Polana, Eulalia and Clarissa have a blue-to-neutral reflectance
spectra, while the other families show a slightly redder reflectance spectra.
This pattern has been already noted previously (Morate et al., 2019). Within
this second group, the Gaia DR3 data of Chaldaea, Primordial, Tamara, Erigone
and perhaps Klio families show evidence of the 700 $\text{\,}\mathrm{nm}$
hydration band.

In addition, the Gaia DR3 spectra of Polana and Eulalia families can be
distinguished in the near-ultraviolet region below 550 $\text{\,}\mathrm{nm}$,
where Eulalia family appears bluer than Polana (Fig. 3). This spectral
behaviour is also evident specifically on asteroids (142) Polana and (495)
Eulalia, the largest members of their respective families (Fig. 3). A similar
behaviour can be found in the ground-based reflectance spectra of asteroids
(142) Polana and (495) Eulalia (de León et al., 2016). In these data, (495)
Eulalia turns bluer than (142) Polana at wavelengths shorter than $\sim$450
$\text{\,}\mathrm{nm}$.

###  4.1 Comparison with ground-based observations

In order to compare our results with previous studies, we downloaded the
PRIMASS-L Spectra Bundle V1.0 from the NASA PDS archive (Pinilla-Alonso et
al., 2021), which contains VIS reflectance spectra of several members of the
C-complex IMB asteroid families. In this dataset, asteroids’ spectra are
already organised by family, with the exception of the Tamara and the
Primordial families that are not listed, while the Eulalia and the Polana
families are merged together. Hence, we separated Eulalia and Polana families’
members on the basis of their position in the $(a,1/D)$ space, namely, objects
with $1/D>-1.7(a-2.49)$ and $p_{V}<0.12$ were assigned to the Eulalia family,
while objects with $1/D<-1.7(a-2.49)$ and $p_{V}<0.12$ were assigned to the
Polana family. Finally, for all families, we removed interlopers as reported
in their respective publications, as well as the spectra that are classified
as D and T types. Figure 2 shows that the agreement between Gaia DR3 and
PRIMASS-L reflectance spectra is in general quite good, with the exception of
the Svea family, where the majority of PRIMASS-L spectra have a blue slope,
while Gaia DR3 one is redder. One of the possible reason for this discrepancy
is that there are only two objects of the Svea family with DR3 reflectance
spectra that pass our filters, namely the asteroids (329) and (102626). The
former, which is the asteroid Svea itself, has also a red sloped reflectance
spectrum in PRIMASS-L, while for the latter there are no literature spectra
previous to Gaia DR3.

Figure 2: Comparison of Gaia DR3 average spectra (red line) for each IMB
primitive family to the ground-based visible spectra of the literature (gray
lines) from the PRIMASS-L survey (see text). Gaia DR3 reflectance spectra are
plotted with and without the correction at the blue most wavelengths of
Tinaut-Ruano et al. (2023).

We also used a classical $\chi^{2}$ technique (e.g., Avdellidou et al., 2022)
to measure the goodness of fit between the Gaia DR3 and the PRIMASS-L
reflectance spectra for the asteroid families of Fig. 2. First, for each
family, $f$, and for each PRIMASS-L reflectance spectrum, we determined a
fitting natural spline (Python csaps) in order to evaluate PRIMASS-L
reflectances at the same wavelengths $\lambda$ of the Gaia DR3 ones. Next, for
each family, we calculated a PRIMASS-L mean spectrum $\overline{R}_{f}$ and
its standard deviation $\overline{\sigma}_{f}$. Finally, we calculated a
$\chi^{2}_{f}=\sum_{\lambda}\left[\left(\overline{R}_{f}(\lambda)-R_{f}(\lambda)\right)^{2}/\overline{\sigma}_{f}^{2}\right]$,
where $R_{f}(\lambda)$ is the Gaia DR3 average reflectance spectrum of the
family $f$. We also calculated the reduced $\chi^{2}$ by the classical
expression $\tilde{\chi}^{2}_{f}=\chi^{2}_{f}/\nu$, where $\nu$ is the number
of degrees of freedom (Press et al., 1992) that in our case is the number of
Gaia DR3 bands used in the expression above minus 1. Results are given in
Table 2. Assuming the $\chi^{2}$ statistic, one can estimate the maximum
$\tilde{\chi}^{2}=1+\sqrt{2\nu}/\nu$ (Press et al., 1992; Hanuš et al., 2018)
within which it is the expected to have 68% of the cases, i.e., the PRIMASS-L
spectra, assuming that spectral variability is random. Since
$1+\sqrt{2\nu}/\nu$ is equal to 1.5 and 1.4 for the cases with $\nu$ equal to
8 and 12, we can deduce that the Gaia DR3 reflectance spectra fits well those
from PRIMASS-L for all the families that the surveys have in common.

Table 2: $\chi^{2}$-values between Gaia DR3 and PRIMASS-L reflectance spectra
for the low-albedo asteroid families of the inner main belt.

Family | $\chi^{2}_{f}$ | $\nu$ | $\tilde{\chi}^{2}_{f}$
---|---|---|---
Chaldaea | 1.69 | 8 | 0.21
Chimaera | 7.17 | 8 | 0.90
Clarissa | 7.60 | 8 | 0.95
Erigone | 3.12 | 8 | 0.39
Eulalia | 5.40 | 12 | 0.45
Klio | 0.70 | 8 | 0.09
Polana | 7.99 | 12 | 0.67
Primordial | - | - | -
Sulamitis | 1.72 | 8 | 0.22
Svea | 3.23 | 8 | 0.41
Tamara | - | - | -
333There are no spectra for the Primordial and the Tamara families from the
PRIMASS-L of Pinilla-Alonso et al. (2021).

Concerning the distinguishability between the Polana and Eulalia families,
previous works (de León et al., 2016; Pinilla-Alonso et al., 2016) concluded
that members of the two families could not be distinguished on the basis of
visible and/or infrared spectroscopy. Tatsumi et al. (2022) also
spectroscopically compared the Eulalia and the Polana families (and the Themis
family) focusing on the blue (or as they call it the NUV) region (350 – 550
$\text{\,}\mathrm{nm}$) of the spectrum. Namely, these authors obtained
ground-based, low-resolution visible spectra as blue as 350
$\text{\,}\mathrm{nm}$ of few asteroids belonging to these families. They also
used literature spectroscopic data from the eight colour asteroid survey
(ECAS, Zellner et al., 1985). Tatsumi et al. (2022) calculated the NUV and VIS
spectral slopes by straight line least-square fit to the spectra data of each
asteroid between 360 and 550 $\text{\,}\mathrm{nm}$ and 550 and 850
$\text{\,}\mathrm{nm}$, respectively. By comparing the NUV and VIS slopes of
the asteroids from the different families, these authors found no significant
differences between the Polana and Eulalia families.

However, using the same data from Tatsumi et al. (2022), we reached a
different conclusion: We used the two-dimensional Kolmogorov–Smirnov test (K–S
test) to verify the null hypothesis that the NUV and VIS slopes from the TNG
observations reported in Table 4 of Tatsumi et al. (2022) from Eulalia and
Polana could come from the same distribution. We found that this null
hypothesis could be rejected at 97.2% confidence level with a K-S test
distance of 0.71. Also, we performed the K-S test with a Monte Carlo method
between the nominal and random draws from the same data of the asteroids of
the Polana family and found that, in 106 Monte Carlo iterations, the max K-S
test distance obtained with random errors is 0.625: i.e., we never obtained
0.71. This means that there is a probability smaller than 10-6 to be able to
obtain Eulalia NUV and VIS spectral slopes from random errors from the Polana
NUV and VIS slopes. If we add the NUV and VIS spectral slopes from Table 4 of
Tatsumi et al. (2022) of the ECAS survey, the difference between the asteroids
from the Polana and the Eulalia families decreases, but still the two families
remain distinguishable. In particular, the K-S test still indicates that the
two samples from the Eulalia and from the Polana families are not obtained
from the same distribution with a confidence level of 95% to reject the null
hypothesis. We concluded that the data from the work of Tatsumi et al. (2022)
do show that the Polana and Eulalia families can be distinguished, contrary to
what was stated by the authors. Independent evidence of the distinction
between the colours of the Polana and Eulalia family members is also claimed
in a recent work based on the Sloan Digital Sky Survey spectrophometry of
asteroids (McClure & Emery, 2022).

###  4.2 The source region of the NEAs Bennu and Ryugu

Previous dynamical and spectroscopic considerations lead to the conclusion
that Bennu originated from the low albedo component of what was, once upon a
time called the Nysa-Polana clan (Campins et al., 2010). It was later shown
that this low albedo component is actually composed by two dynamically
distinguishable families named Polana (or New Polana) and Eulalia (Walsh et
al., 2013). Further investigations assessed that Bennu has about 70 and 30%
probability to originate from the Polana and Eulalia families, respectively
(Bottke et al., 2015).

Figure 3 shows the comparison between the reflectance spectra of Eulalia and
Polana families and the reflectance spectra of Bennu obtained by the OSIRIS-
REx Visible and InfraRed Spectrometer (OVIRS) and MapCam instruments on board
OSIRIS-REx (DellaGiustina et al., 2020). In the ultraviolet region, it appears
that Bennu is marginally more similar to the Eulalia family members than those
of the Polana family: namely, Gaia DR3 results slightly favour an origin of
Bennu from the Eulalia family because the trend of Bennu’s reflectance into
the wavelength region bluer than 450 $\text{\,}\mathrm{nm}$ is more consistent
with the reflectance of the Eulalia than of the Polana family (Fig. 3).
Likewise, Bennu’s reflectance trend into the wavelength region bluer than 450
$\text{\,}\mathrm{nm}$ is more similar to asteroid (495) Eulalia than to the
asteroid (142) Polana.

On the other hand, the average reflectance spectra of the Eulalia and Polana
families are redder than the reflectance spectrum of Bennu longward $\sim$800
$\text{\,}\mathrm{nm}$. It cannot be established with the present Gaia DR3
data whether this is a real effect or an artefact. Indeed, it has been
observed (Gaia Collaboration et al., 2023; Galinier et al., 2023) that Gaia
DR3 asteroid spectra tend to have reflectance values higher than the spectra
of the same objects collected from ground-based telescopes at wavelengths
roughly beyond $\sim$800–900 $\text{\,}\mathrm{nm}$.

Figure 3 subplot also shows that the reflectance spectra of the parent bodies
of each respective families are consistent with the families general trend.

Figure 3: Comparison of the average reflectance spectra of the Eulalia and
Polana family from Gaia DR3 and Bennu from DellaGiustina et al. (2020). The
subplot also shows the Gaia DR3 spectra of (142) Polana and (495) Eulalia.

We also performed a $\chi^{2}$ analysis between the reflectance spectra of
Bennu, obtained by the OSIRIS-REx mission (Fig. 3), and the average
reflectance of the families from the Gaia DR3. Namely, we constructed a mean
reflectance spectrum $R_{B}(\lambda)$ and its standard deviation from
smoothing spline representations (Python csasp) of the reflectance spectra
from OVIRS and MapCam (DellaGiustina et al., 2020). Next, we calculated the
reduced $\chi^{2}$,
$\tilde{\chi}^{2}_{B,f}=(1/\nu)\sum_{\lambda}\left[\left(R_{B}(\lambda)-R_{f}(\lambda)\right)^{2}/\overline{\sigma}_{f}^{2}\right]$,
where $R_{f}(\lambda)$ is the Gaia DR3 average reflectance spectrum of the
family $f$, as before, and $\nu$ is the degrees of freedom, i.e., the number
of wavelength datapoints - 1. The lower the value of $\tilde{\chi}^{2}_{B,f}$,
the better the spectroscopic match is. Firstly, we considered all data points
in the wavelength range between 450 and 950 $\text{\,}\mathrm{nm}$, as shown
in Fig. 3 (OVIRS and MapCam spectra do not cover wavelengths bluer than 450
$\text{\,}\mathrm{nm}$). We found $\tilde{\chi}^{2}_{B,f}$-values of 3.5 and
6.0 for the Eulalia and the Polana families, respectively; all other families
producing $\tilde{\chi}^{2}_{B,f}$-values $>$ 76\. This result further
indicates that the Eulalia is the first and Polana is the second family more
spectroscopic alike Bennu. Given that the spectroscopic match between Eulalia
(and Polana) and Bennu, is visually better in the bluer region of the spectrum
(Fig. 3), we also calculated $\tilde{\chi}^{2}_{B,f}$ in different wavelength
ranges obtained from the aforementioned one by reducing the upper bound from
950 to 800 $\text{\,}\mathrm{nm}$. We found that the values of
$\tilde{\chi}^{2}_{B,f}$ decrease monotonically as the upper bound decreases,
obtaining values below 1 for the Eulalia family when the upper wavelength
bound is reduced to 800 $\text{\,}\mathrm{nm}$. The corresponding values are
reported in Table 3. We also found that Polana is the family always providing
the second best match. It is common to accept reduced $\chi^{2}$ values
$<(1+\sqrt{2\nu}/\nu)$ (Press et al., 1992): only the Eulalia family satisfies
this constraint.

Table 3: Reduced $\chi^{2}$-values between Gaia DR3 average reflectance spectra of the low-albedo asteroid families of the inner main belt and the average reflectance spectrum of Bennu. The $\chi^{2}$-values are calculated here on the wavelength range between 450 and 800 $\text{\,}\mathrm{nm}$. The table is sorted by the $\chi^{2}$-value, from the smallest to the largest. Family | $\tilde{\chi}^{2}_{B,f}$ | $\nu$
---|---|---
Eulalia | 0.99 | 6
Polana | 3.34 | 6
Clarissa | 10.00 | 6
Tamara | 25.33 | 6
Erigone | 68.13 | 6
Chaldaea | 88.03 | 6
Klio | 105.47 | 6
Sulamitis | 124.81 | 6
Chimaera | 752.72 | 6
Primordial | 791.36 | 6
Svea | 1540.66 | 6
Table 4: Reduced $\chi^{2}$-values between Gaia DR3 average reflectance spectra of the low-albedo asteroid families of the inner main belt and the average reflectance spectra of Ryugu. The $\chi^{2}$-values are calculated here on the wavelength range between 370 and 950 $\text{\,}\mathrm{nm}$. The table is sorted by the $\chi^{2}$-value, from the smallest to the largest.  Family | $\tilde{\chi}^{2}_{R,f}$ | $\nu$
---|---|---
Polana | 1.91 | 12
Clarissa | 3.30 | 12
Eulalia | 3.37 | 12
Tamara | 3.79 | 12
Klio | 5.72 | 12
Erigone | 5.75 | 12
Chaldaea | 6.72 | 12
Sulamitis | 12.17 | 12
Primordial | 25.33 | 12
Chimaera | 117.90 | 12
Svea | 606.61 | 12

Figure 4 shows the comparison between the reflectance spectra of Eulalia and
Polana families and the reflectance spectra of Ryugu obtained by the Optical
Navigation Camera (ONC) instrument on board the Hayabusa2 (Sugita et al.,
2019), along with ground-based spectra (Moskovitz et al., 2013; Perna et al.,
2017). In the blue and ultraviolet regions, Ryugu shows a reflectance spectra
that are, in principle, compatible with both Polana and Eulalia families
average spectra. On the other hand, at wavelengths longer than 600
$\text{\,}\mathrm{nm}$, the average Gaia DR3 reflectance spectra of Polana and
Eulalia families have somewhat lower reflectance values compared to the Ryugu
spectra measured from Hayabusa2. However, they appear still marginally within
the error-bars of Ryugu’s ground-based spectra.

Figure 4: Comparison of the average reflectance spectra of the Eulalia and
Polana family from Gaia DR3 and Ryugu from Sugita et al. (2019), Moskovitz et
al. (2013), and Perna et al. (2017). The subplot also shows the Gaia DR3
spectra of (142) Polana and (495) Eulalia.

As for Bennu, we performed a $\chi^{2}$ analysis between the mean reflectance
spectra of Ryugu, $R_{R}(\lambda)$, calculated from averaging smoothing spline
fits to the spectra obtained by the Hayabusa2 mission and ground based
telescopes (Fig. 4), and the average reflectance spectra of each family, $f$,
from the Gaia DR3, $R_{f}(\lambda)$: namely,
$\tilde{\chi}^{2}_{R,f}\leavevmode\nobreak\ =\leavevmode\nobreak\
(1/\nu)\sum_{\lambda}\left[\left(R_{R}(\lambda)-R_{f}(\lambda)\right)^{2}/\overline{\sigma}_{f}^{2}\right]$.
Considering all the data points in the wavelength range between 370 and 950
$\text{\,}\mathrm{nm}$, we obtained the $\tilde{\chi}^{2}_{R,f}$-values
reported in Table 4 . The family with the spectrum producing the minimum
$\chi^{2}$ value is Polana, followed by the Clarissa and the Eulalia families.
However, this reduced $\chi^{2}$ value is never close to 1 (the lowest value
is $\sim$1.9) for the different spectral ranges here considered, indicating
that this spectral match is never formally satisfactory. Contrary to the case
of Bennu, calculating the $\tilde{\chi}^{2}_{R,f}$ in shorter wavelength
intervals, such as between 450 and 800 $\text{\,}\mathrm{nm}$, does not lower
the minimum $\tilde{\chi}^{2}_{R,f}$-value.

##  5 Conclusions

Using the unprecedented Gaia DR3 sample of visible asteroid spectra, we
created the average reflectance spectra of the 11 dark primitive asteroid
families of the IMB that belong to the spectroscopic C-complex.

In this work, we reported on their similarities, but also their differences,
and we compared nine of them with earlier ground-based spectroscopic results.
Gaia DR3 provides invaluable reflectance information between 370 and 500
$\text{\,}\mathrm{nm}$, a region that is poorly studied so far from the
ground, but can indicate differences among the C-complex population.

Consistently with previous results, we found that there appear to be two
spectroscopic groups within these families: Eulalia, Polana and Clarissa are
spectroscopically bluer than all the others. Within the redder groups of
families, Erigone, Tamara, Chaldaea and the Primordial, appear to have a 700
$\text{\,}\mathrm{nm}$ absorption band, indicative of hydration.

We found that the Eulalia and Polana adjacent families can be
spectroscopically distinguished in the wavelength region between 370 to 500
$\text{\,}\mathrm{nm}$, where Eulalia is bluer than Polana.

The comparison of the average family spectra with those of the NEAs Bennu and
Ryugu showed that these asteroids are spectroscopically more similar to
Eulalia and Polana, respectively, than to other families.

In particular, we found that the Gaia DR3 average spectrum of the Eulalia
family is a good spectroscopic match for Bennu’s spectrum in the wavelength
range between 450 and 800 $\text{\,}\mathrm{nm}$. On the other hand, Polana is
the family which average DR3 spectrum has the smallest reduced $\chi^{2}$ when
compared to the average spectrum of Ryugu.

###### Acknowledgements.

We acknowledge support from the ANR ORIGINS (ANR-18-CE31-0014). This work has
made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This work is based on data
provided by the Minor Planet Physical Properties Catalogue (MP3C) of the
Observatoire de la Côte d’Azur.

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