# Spectral evolution of dark asteroid surfaces induced by space weathering
over a decade

Sunao Hasegawa Institute of Space and Astronautical Science, Japan Aerospace
Exploration Agency, 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210,
Japan  Francesca E. DeMeo Department of Earth, Atmospheric and Planetary
Sciences, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA  Michaël
Marsset Department of Earth, Atmospheric and Planetary Sciences, MIT, 77
Massachusetts Avenue, Cambridge, MA 02139, USA  European Southern Observatory
(ESO), Alonso de Córdova 3107, 1900 Casilla Vitacura, Santiago, Chile  Josef
Hanuš Charles University, Faculty of Mathematics and Physics, Institute of
Astronomy, V Holešovičkách 2, CZ-18000, Prague 8, Czech Republic  Chrysa
Avdellidou Université Côte d’Azur, CNRS–Lagrange, Observatoire de la Côte
d’Azur, CS 34229 – F 06304 NICE Cedex 4, France  Marco Delbo Université Côte
d’Azur, CNRS–Lagrange, Observatoire de la Côte d’Azur, CS 34229 – F 06304 NICE
Cedex 4, France  Schelte J. Bus Institute for Astronomy, University of Hawaii,
2860 Woodlawn Drive, Honolulu, HI 96822-1839, USA  Hidekazu Hanayama
Ishigakijima Astronomical Observatory, National Astronomical Observatory of
Japan, 1024-1 Arakawa, Ishigaki, Okinawa 907-0024, Japan  Takashi Horiuchi
Institute of Astronomy, the University of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo
181-0015, Japan  Driss Takir Jacobs, NASA Johnson Space Center, Houston, TX
77058, USA  Emmanuël Jehin Space sciences, Technologies and Astrophysics
Research Institute, Université de Liege, allée du 6 Août 17, 4000 Liège,
Belgium  Marin Ferrais Aix Marseille Université, CNRS, CNES, Laboratoire
d’Astrophysique de Marseille, Marseille, France  Jooyeon Geem Department of
Physics and Astronomy, Seoul National University, Gwanak-gu, Seoul 08826,
Republic of Korea  SNU Astronomy Research Center, Seoul National University,
Gwanak-gu, Seoul 08826, Republic of Korea  Myungshin Im Department of Physics
and Astronomy, Seoul National University, Gwanak-gu, Seoul 08826, Republic of
Korea  SNU Astronomy Research Center, Seoul National University, Gwanak-gu,
Seoul 08826, Republic of Korea  Jinguk Seo  Department of Physics and
Astronomy, Seoul National University, Gwanak-gu, Seoul 08826, Republic of
Korea  SNU Astronomy Research Center, Seoul National University, Gwanak-gu,
Seoul 08826, Republic of Korea  Yoonsoo P. Bach Department of Physics and
Astronomy, Seoul National University, Gwanak-gu, Seoul 08826, Republic of
Korea  SNU Astronomy Research Center, Seoul National University, Gwanak-gu,
Seoul 08826, Republic of Korea  Sunho Jin Department of Physics and Astronomy,
Seoul National University, Gwanak-gu, Seoul 08826, Republic of Korea  SNU
Astronomy Research Center, Seoul National University, Gwanak-gu, Seoul 08826,
Republic of Korea  Masateru Ishiguro Department of Physics and Astronomy,
Seoul National University, Gwanak-gu, Seoul 08826, Republic of Korea  SNU
Astronomy Research Center, Seoul National University, Gwanak-gu, Seoul 08826,
Republic of Korea  Daisuke Kuroda Japan Spaceguard Association, Bisei
Spaceguard Center 1716-3 Okura, Bisei, Ibara, Okayama 714-1411, Japan  Richard
P. Binzel Department of Earth, Atmospheric and Planetary Sciences, MIT, 77
Massachusetts Avenue, Cambridge, MA 02139, USA  Akiko M. Nakamura Department
of Planetology, Graduate School of Science, Kobe University, 1-1 Rokkodai-cho,
Nada-ku, Kobe 657-8501, Japan  Bin Yang Núcleo de Astronomía, Facultad de
Ingenieríay Ciencias, Universidad Diego Portales, Chile  Pierre Vernazza Aix
Marseille Université, CNRS, CNES, Laboratoire d’Astrophysique de Marseille,
Marseille, France

###### Abstract

The surface of airless bodies like asteroids in the Solar System are known to
be affected by space weathering. Experiments simulating space weathering are
essential for studying the effects of this process on meteorite samples, but
the problem is that the time spent to reproduce space weathering in these
experiments is billions of times shorter than the actual phenomenon. In
December 2010, the T-type asteroid 596 Scheila underwent a collision with a
few-tens-of-meters impactor. A decade later, there is an opportunity to study
how the surface layer of this asteroid is being altered by space weathering
after the impact. To do so, we performed visible spectrophotometric and near-
infrared spectroscopic observations of 596 Scheila. The acquired spectrum is
consistent with those observed shortly after the 2010 impact event within the
observational uncertainty range. This indicates that the surface color of dark
asteroids is not noticeably changed by space weathering over a 10-year period.
This study is the first to investigate color changes due to space weathering
on an actual asteroid surface in the Solar System. Considering that fresh
layers are regularly created on asteroid surfaces by collisions, we suggest a
genetic link between D/T-type and dark (low albedo) X-complex asteroids and
very red objects such as 269 Justitia, 732 Tjilaki (and 203 Pompeja). New
observations show that 203 Pompeja has a X-type-like surface, with some local
surface areas exhibiting a very red spectrum.

Small Solar System bodies(1469) — Asteroids (72) — Main belt asteroids(2036) —
Asteroid surfaces (2209)

††journal: ApJ††facilities: IRTF:3.2m, Murikabushi:1.05m, SAO:1.0m, TRAPPIST-
South:0.6m, ZTF:1.2m††software: IRAF (Tody, 1993), IDL

##  1 Introduction

It is known that the surface layers of airless bodies in the Solar System are
usually not fresh, having been exposed to space for long periods of time. The
effect responsible for altering the surfaces of these objects and affecting
their colors and reflectance spectra is called space weathering. This process
was first evidenced by the distinct colors of the youngest craters on the Moon
(Gold, 1955). Analysis of lunar soil samples then revealed that space
weathering on the lunar surface is caused by the impact of micrometeoroids and
implantation of solar wind (Hapke et al., 1975).

The lack of main-belt asteroids with spectra similar to those of ordinary
chondrites suggested that space weathering also affected the surface layers of
S-complex asteroids (e.g., Pieters & McFadden, 1994). Telescopic observations
of S-complex and Q-type asteroids (Binzel et al., 1996) and rendezvous
observations of S-complex asteroids by spacecraft (e.g., Veverka et al., 2000;
Hiroi et al., 2006) later confirmed that space weathering causes changes in
the reflectance spectra of the surfaces of S-complex asteroids. In particular,
the samples returned from the S-complex asteroid 25143 Itokawa were
compositionally similar to ordinary chondrites (e.g., Nakamura et al., 2011;
Yurimoto et al., 2011). Nanophase iron particles, which had been found on the
lunar soil (Keller & McKay, 1993), were also found in the Itokawa samples
(Noguchi et al., 2011), thereby proving that the reflectance spectra of
S-complex asteroids are reddened by space weathering.

For a long time, space weathering of dark asteroids (albedo values less than
0.1) such as C-complex and D-type asteroids did not receive much attention
because many dark asteroids had been found to exhibit spectra consistent with
carbonaceous chondrites (e.g., Clark et al., 2002). Nesvorný et al. (2005) was
the first to show a trend in space weathering of C-complex asteroids based on
correlations between family ages and spectral slopes. Lantz et al. (2018)
further showed trends in spectral slope due to space weathering for each
spectral type of dark asteroids observed from telescopic and laboratory study
data.

Laboratory simulation experiments of space weathering using minerals and
meteorites are very important for understanding spectral changes in response
to this process on the surface of asteroids (e.g., Moroz et al., 1996; Yamada
et al., 1999; Dukes et al., 1999). In particular, the results of simulated
space weathering experiments on carbonaceous chondrites, which are considered
to be the meteorite counterparts of dark asteroids, are essential to
understand the spectral changes induced by space weathering on this type of
object (e.g., Vernazza et al., 2013; Matsuoka et al., 2015; Lantz et al.,
2017). While laboratory experiments are important research methods for
understanding space weathering, there is one obstacle that cannot be overcome:
time simulated experiment alter the spectra of the exposed samples over milli-
year timescales, whereas the spectral changes induced by space weathering on
actual asteroids act over thousand-year timescales or more. Although the
irradiating fluxes are increased billions of times in intensity with respect
to actual space weathering during the experiments, it cannot be guaranteed
that this properly reproduces what is happening in nature over such long
timescales. If experiments could be performed on meteorites such that they
match the space-weathered age of the actual observed asteroid and if the
resulting irradiated meteorite and the asteroid spectra could be compared,
this problem would be resolved.

Another point of interest is the characteristic time over which space
weathering significantly affects the spectra. Studies combining asteroid
spectra with the age of their respective collisional families showed spectral
changes in response to space weathering over million- to billion-year
timescales (Nesvorný et al., 2005; Vernazza et al., 2009). In addition, space
weathering has been reported to be efficient over thousand year timescales,
although this was reported for ordinary chondrites instead of carbonaceous
chondrites (Noguchi et al., 2014; Jin & Ishiguro, in press). However, it
remains unknown how spectra change due to space weathering over shorter time
periods. In particular, it would be very interesting to know if spectral slope
changes linearly or logarithmically with time over periods shorter than a
thousand to a million years.

The surface of the T-type asteroid 596 Scheila with an albedo of 0.038111Mean
values obtained from IRAS (Tedesco et al., 2002), AKARI (Usui et al., 2011),
and WISE (Masiero et al., 2011). was refreshed during an impact with a few-
tens-of-meters projectile in 2010 (Ishiguro et al., 2011a, b). In this
collision event, the spectral type of 596 turned redder, from T type to D type
(Hasegawa et al., 2022). Assuming that the surface layer of 596 was space-
weathered before the impact, and that a fresh surface appeared after the
event, it was concluded that objects compositionally similar to 596 become
bluer in spectral slope due to space weathering. This year marks approximately
10 years since the collisional event. This means that the refreshed surface of
596 has been exposed to space weathering for a period of approximately 10
years. To test whether the surface layer of 596 and its reflectance spectrum
changed during that period of time, we performed spectral observations of 596
in 2022, more than a decade after the collisional event.

In this study, we investigate the spectral changes caused by space weathering
over a short timescale of 10 years, which is a much shorter timescale than the
formation age of the solar system. A comparable study of asteroids over a
short timescale of 300 years was recently reported by Fatka et al. (2022).
These authors found that two near-Earth asteroids (NEAs) 2019
${\mathrm{PR_{2}}}$ and 2019 ${\mathrm{QR_{6}}}$ may have formed as a pair
about three hundred years ago. The authors also showed that the two asteroids
have spectral slopes that are equal to or redder than those of D-type
asteroids. This suggests that the fresh surface layers of D-type asteroids
before space weathering effects become apparent may be similar to very red
objects (VROs) found by Hasegawa et al. (2021), which have a redder spectral
slope than the average D-type asteroid (DeMeo et al., 2009; Hasegawa et al.,
2022). Figure 1 shows that the very red NEAs, which may be very young, have a
similar slope in visible spectra to the VROs 203 Pompeja with an albedo of
0.0451 and 269 Justitia with an albedo of 0.0801. This indicates that the
surface ages of 203 and 269 may be as young as a few hundred years. Therefore,
we also conducted observations of 203 and 269 to investigate if a slope change
occurred in their spectra over time. Note that one further VRO: 732 Tjilaki
with an albedo of 0.068222Mean values obtained from IRAS (Tedesco et al.,
2002) and AKARI (Usui et al., 2011). was discovered in Bourdelle de Micas et
al. (in press). We also investigated if a slope change of occurred in 732
spectra over time using literature.

Figure 1:  Comparison of the visible spectra of VROs: 203 Pompeja, 269
Justitia, 2019 ${\mathrm{PR_{2}}}$, and 2019 ${\mathrm{QR_{6}}}$. The figure
shows the spectra in the 0.4–1.0 $\mu$m wavelength regions, normalized to 1 at
0.55 $\mu$m.

##  2 Observations and Data Analysis

New near-infrared (NIR) observations of 596 Scheila were carried out with the
SpeX NIR spectrograph (Rayner et al., 2003) on the 3.2 m NASA Infrared
Telescope Facility (IRTF) located on Maunakea, Hawaii on 2022 April 4. Both
Landolt 102-1081 and Landolt 107-998 were used as solar analogues for the NIR
observation (Marsset et al., 2020). Reduction of NIR spectral images was
conducted using the Image Reduction and Analysis Facility (IRAF; Tody 1993)
and the Interactive Data Language (IDL). Close in time, visible observations
were conducted in the synchronous imaging camera with $g^{\prime}$-, $R_{\rm
C}$-, and $I_{\rm C}$-bands (Yanagisawa et al., 2010) on the 1.05 m
Murikabushi Telescope located on Ishigakijima, Japan on 2022 Febrary 8.
HD142801 was used as a calibration star for the visible observation. Data
reduction for photometry was performed using the IRAF. The procedures for the
NIR and visible observations and reduction are as described in Binzel et al.
(2019) and Hasegawa et al. (2021), respectively.

NIR spectrum of 269 Justitia was collected in 2009 June 15 using SpeX/IRTF
(Emery private comm.). HD 124901 was used as a solar analogue star. Spectral
images were reduced by the IDL. The procedures for the NIR observations and
reduction of 269 are as described in Emery et al. (2011).

NIR spectra of 203 Pompeja were obtained in 2020 December 1 and 2022 June 10
and 27. As in previous observations, the observations were made using
SpeX/IRTF. BD+00 2717, HD 249566, and HD 106116 were used as a solar analogue
star on 2020 December 1, 2022 June 10, and 2022 June 27, respectively.
Reduction of the spectroscopic images was performed with the IDL. The
procedures for the NIR observations and reduction of 203 are as described in
Takir & Emery (2012) and Avdellidou et al. (2021). Visible spectra of 203
Pompeja were obtained on 2022 June 18, 19, 29, and 30. Colorimetric
observations were acquired with the robotic telescope TRAPPIST-South (TS)
(Jehin et al., 2011) located, at the ESO La Silla Observatory in Chile. TS is
a 0.6 m Ritchey-Chrétien telescope operating at f/8 and equipped with a
thermoelectrically-cooled FLI ProLine 3041-BB CCD camera. The data were
reduced using the PHOTOMETRYPIPELINE (Mommert, 2017) and calibrated to the
band they were observed using the Pan-STARRS catalogue.

##  3 596 Scheila

###  3.1 Spectroscopic results

We compared the spectra of 596 Scheila obtained before, immediately after the
2010 impact, and a decade after the event to search for any spectral change
induced by space weathering. Spectral data before the 2010 impact event are
from Bus & Binzel (2002) and DeMeo et al. (2009). NIR spectral data obtained
immediately after the impact event were retrieved from Yang & Hsieh (2011) and
Hasegawa et al. (2022). Visible spectrophotometric data immediately after
impact were combined from Trigo-Rodriguez et al. (2011), Betzler et al.
(2012), Hsieh et al. (2012), Jewitt (2012), and Shevchenko et al. (2016). In
addition to the data acquired in the present work, multi-filter photometric
data from Sergeyev et al. (2022) were used as part of the “10 years after the
event” dataset as those data were obtained on 2018 August 20. As reference
data, visible reflectance spectrum from Gaia is an average of several
observations taken somewhere between 2014 August 5 and 2017 May 28, i.e.,
$\sim$3.5 and $\sim$6.5 years after the impact event are also shown (Gaia
Collaboration et al., in press)333The epoch spectra are not available in the
Data Release 3 (DR3). All the DR3 asteroid spectra are averages of epoch
spectra taken between 2014 August 5 and 2017 May 28.. Table 1 indicates the
sub-observer coordinates of NIR observations before, immediately after, and a
decade after the impact, calculated based on the shape model of 596 from Hanuš
et al. (2021). Although the observed latitude and longitude of each
observation is different, Hasegawa et al. (2022) indicated that the ejecta
discharged by the 2010 impact event was deposited over the entire surface of
the asteroid and that the surface layer was homogeneously covered by fresh
material.

Figure 2 compares the reflectance spectrum of 596 before, immediately after,
and 10 years after the impact event. The spectral slope of 596 over 0.8–2.5
$\mu$m before, immediately after, and a decade after the impact event is 25.5,
46.7(40.5), and 42.2% $\mu$m-1, respectively. The slopes of the NIR spectra
immediately after the impact and about 10 years after the impact are
consistent within the expected slope uncertainty of 4.2% $\mu$m-1 for SpeX
measurements (Marsset et al., 2020). The slope consistency between the visible
spectra of 596 obtained over all observed periods indicates no significant
change over the decade at these wavelengths within the observational
uncertainty range.

Table 1: Sub-observer coordinates of 596 Scheila during NIR observations based
on Hanuš et al. (2021)

Observation time | Timing of | Sub-Earth point | Spectral slope | $R_{\rm h}^{c}$ | $\Delta^{c}$ | $\alpha^{c}$ | Reference
---|---|---|---|---|---|---|---
| Observation |  ${\lambda_{1}}^{a}$ [°] |  ${\phi_{1}}^{a}$ [°] |  ${\lambda_{2}}^{b}$ [°] |  ${\phi_{2}}^{b}$ [°] | [% $\mu$m-1] | [au] | [au] | [°] |
2002/06/01 09:40 | before impact | 137 | 46 | 232 |  $-$47 | 25.5 | 2.45 | 1.43 | 0.4 | DeMeo et al. (2009)
2011/01/04 14:50 | immediately after impact | 132 |  $-$39 | 263 | 43 | $40.5^{d}$ | 3.07 | 2.26 | 12.0 | Yang & Hsieh (2011)
2011/01/05 14:20 | immediately after impact | 319 |  $-$39 | 90 | 44 | $40.5^{d}$ | 3.07 | 2.25 | 11.7 | Yang & Hsieh (2011)
2011/02/07 09:31 | immediately after impact | 99 |  $-$40 | 225 | 50 | 46.7 | 3.02 | 2.07 | 6.4 | Hasegawa et al. (2022)
2022/04/09 14:29 | a decade after impact | 357 | 51 | 179 |  $-$42 | 42.2 | 2.46 | 1.73 | 19.2 | This study

Note. —

aPole solution 1: ${\lambda}_{1}$ = 110 [°], ${\beta}_{1}$ = $-$24 [°], Period
= 15.8605 [hr]

bPole solution 2: ${\lambda}_{2}$ = 273 [°], ${\beta}_{2}$ = $-$38 [°], Period
= 15.8609 [hr]

cThe heliocentric ($R_{\rm h}$) and geocentric distances ($\Delta$) and phase
angle ($\alpha$) for asteroid observations were taken from the NASA/ Jet
Propulsion Laboratory (JPL) HORIZONS ephemeris generator systeme. dThe value
of the combined spectrum in Yang & Hsieh (2011). e
https://ssd.jpl.nasa.gov/horizons

Figure 2:  Comparison of 596 Scheila’s NIR spectra before, immediately after,
and a decade after the 2010 impact event. The figure shows spectra of 596 in
the 0.4–2.5 $\mu$m wavelength region, normalized to 1 at 0.55 $\mu$m. The
black arrows indicate the slope uncertainty from Marsset et al. (2020)
centered at the reflectance value of the spectra at 2.40 $\mu$m, as shown by
the black dashed lines.

###  3.2 Spectral changes due to phase angle

Regarding possible changes in the spectral gradient, it was shown in Hasegawa
et al. (2022) that particle size effect can not explain the observed spectral
reddening changes. However, the possibility of a change in the slope of the
spectrum due to a change in roughness and phase angle was not discussed in
Hasegawa et al. (2022) and is discussed below.

Binzel et al. (2015) pointed out that spectral changes occur due to surface
roughness effects in case of regolith-free surfaces. However, the surface
layer of 596 is considered to be covered with regolith (Ishiguro et al.,
2011a; Yang & Hsieh, 2011). Therefore, changes in spectral slope due to
roughness do not need to be taken into account in the case of 596.

Sanchez et al. (2012) described the spectral dependence on the phase angle of
ordinary chondrite. For ordinary chondrites, changes in spectral slope are not
seen below a phase angle of 30 degrees (Figure 3). Considering the
observational slope uncertainty (Marsset et al., 2020), unless the
observations are repeated several times, in which case the uncertainty on the
mean can be reduced (Figure 3). Thomas et al. (2014) investigated the spectral
dependence on the phase angle of S-type asteroids; 433 Eros, 1036 Ganymed, and
1627 Ivar. Among these objects, 433 Eros was shown to be spectrally
homogeneous across its surface by NASA’s spacecraft NEAR Shoemaker (Bell et
al., 2002). No change in spectral slope of 433 is observed up to within 22
degrees of the phase angle (Figure 3). Given the uncertainty of the slope of
the observations shown in Marsset et al. (2020), unless there are multiple
observations, in which case the uncertainty on the mean can be reduced (Figure
3).

While there are examples of phase angle dependence for S-type asteroids as
described above, no studies have investigated the phase-angle dependency of
spectral slope of individual D-type asteroids. However, there are examples
where spectra of various D-type asteroids have been collected to investigate
the dependence of the spectral slope on the phase angle in the visible to
near-infrared wavelength range (Lantz et al., 2018). D-type asteroids have
been shown to have an increasing spectral slope with increasing phase angle,
but, given the uncertainty in the slope of the observations presented in
Marsset et al. (2020), it is not possible to state the change in the slope of
the spectrum within a phase angle between $\sim$2 and $\sim$19 degrees
considering the number of available measurements (Figure 3). However, this
dataset used data from different D-type asteroids, which may have different
spectral slopes at a given phase angle. Hiroi & Hasegawa (2003) studied the
spectral slope of the Tagish lake meteorite by varying the phase angle. The
slope of the spectrum does not change up to a phase angle of 30 degrees, but
the slope of the spectrum increases slightly at a phase angle of 60 degrees
(Figure 3). However, up to within $\sim$50 degrees of the phase angle, the
change is within the range of observation uncertainty. Therefore, 596 having a
spectrum comparable to that of the Tagish Lake meteorite one cannot expect to
detect a slope change at phase angles below $\sim$50 degrees.

NIR observations of 596 before and after the 2010 impact cover a range of
phase angles comprised between 0.4 to 19.2 degrees. Considering that the
spectral slope of 596 varies beyond about three times the observation
uncertainty, it can be concluded that the spectral changes are not due to
changes caused by the phase angle.

Figure 3: Comparison between the spectral slope of Dhurmsala (LL6) and Tagish
Lake (C2) meteorites and Sw-type asteroid 433 Eros and D-type asteroids as
function of phase angle. The D-type asteroid data set is a combination of
multiple D-type asteroids. Gray and yellow areas indicate the slope
uncertainty from Marsset et al. (2020) for the meteorites and the asteroids,
respectively. The solid and dashed lines represent polynomial fits for the
meteorites and asteroids, respectively. The D-type dataset includes the post-
impact spectrum of 596.

###  3.3 Constraints on spectral changes due to space weathering

The spectral match between the spectrum of the fresh surface observed
immediately after the 2010 impact event and the space-weathered one observed a
decade later indicates that the spectra of dark asteroids like 596 do not
evolve over a decade timescale due to space weathering. This was predicted
from a number of space weathering experiments performed on carbonaceous
chondrites (e.g., Vernazza et al., 2013; Matsuoka et al., 2015; Lantz et al.,
2017). However, this is the first time that the absence of spectral changes
due to space weathering is observationally confirmed for an actual asteroid
under conditions where the exposure time is precisely known.

Next, we investigate whether the spectral change due to space weathering acts
linearly or logarithmically with time. Following Ishiguro et al. (2011b), we
assume that the impact on 596 occurred on 2010 December 3. Then, considering
that the average collisional lifetime of 596 for impacts of the scale of the
2010 event is about 104 yr (Hasegawa et al., 2022), we assume that this was
the age of 596’s surface just before the impact, when the spectrum of DeMeo et
al. (2009) was acquired. Under these assumptions, we fitted both a linear and
a logarithmic function to the slope values of the spectra from Yang & Hsieh
(2011) (t=0.09 yr after the collision), Hasegawa et al. (2022) (t=0.18 yr),
and DeMeo et al. (2009) (t=104 yr). By doing so, we find that the slope would
have changed by 0.0018% $\mu$m-1 yr-1 assuming a linear evolution and 3.6%
$\mu$m-1 (log(yr))-1 assuming a logarithmic evolution (Figure 4). That is, if
space weathering changes linearly, the expected slope after about 10 years
would be 43.6% $\mu$m-1, while if it changes logarithmically, the expected
slope after about 10 years would be 36.4% $\mu$m-1. The actual slope we
observed after about 10 years is 42.2%, meaning that the change is smaller
than the expected value from the logarithmic function, although 1.5-$\sigma$
consistent considering the expected 4.2 % $\mu$m-1 measurement uncertainty
(Marsset et al., 2020). This suggests that the spectral change slope evolution
due to space weathering during the first decade is not logarithmic with
respect to time, but linear. Therefore, we find that the NIR reflectance
spectra of reddish dark asteroids like 596 likely do not evolve faster than a
logarithmic function, at least during the first 10 years following a
refreshing event.

Figure 4:  Spectral slope for 596 Scheila against space weathering time. Blue
and orange lines correspond to linear and logarithmic regression of the
spectral slope values of DeMeo et al. (2009), Yang & Hsieh (2011), and
Hasegawa et al. (2022), respectively.

##  4 269 Justitia

The visible spectra of 269 Justitia measured on 1994 November 05 and 2010
January 11 were retrieved from Bus & Binzel (2002) and Cellino et al. (2020),
respectively. Most data collected on 2018 July 16 were retrieved from Sergeyev
et al. (2022). The mean visible spectra was also obtained somewhere between
2014 August 05 and 2017 May 28 (Gaia Collaboration et al., in press)3. These
visible spectra of 269 at 0.45–0.85 $\mu$m are consistent with one another
(Figure 5). This indicates that the visible spectrum of 269 at 0.45–0.85
$\mu$m did not change over 22 years. Two NIR spectra of 269 were acquired on
2002 September 28 (Hasegawa et al., 2021) and 2005 May 11 (DeMeo et al.,
2009). Moreover, the NIR spectrum of 269 was obtained on 2009 June 15 (Emery,
private comm.). The slopes of the three NIR data between 0.8–0.95 $\mu$m are
consistent with the slope of the visible data from Sergeyev et al. (2022)
(Figure 5).

The slope of the visible spectrum of 269 between 0.45–0.95 $\mu$m is found to
have remained unchanged for at least about 16 years. Although the observed
wavelengths are different, the results of unchanged spectra are the same for
269 and 596.

Figure 5:  Spectra of 203 Pompeja (top panels), 269 Justitia (bottom left
panel), and 732 Tjilaki (bottom right panel). Square brackets in these panels
indicate the observation dates. The black arrows indicate the slope
uncertainty from Marsset et al. (2020) centered at the reflectance value of
the spectra at 2.40 $\mu$m, as shown by the black dashed lines.

##  5 732 Tjilaki

Visible spectra of 732 Tjilaki were obtained on 1992 March 04 (Xu et al.,
1995), 1997 March 07 (Lazzaro et al., 2004), and 2021 October 01 (Bourdelle de
Micas et al., in press). The mean visible spectra was also acquire between
2014 August 05 and 2017 May 28 (Gaia Collaboration et al., in press)3. NIR
spectrum was taken on 2021 October 23 were retrieved from Bourdelle de Micas
et al. (in press). These visible spectra of 732 at 0.6–0.9 $\mu$m are
consistent with one another (Figure 5).

The slope of the visible spectrum between 0.6–0.9 $\mu$m of 732 has not
changed for at least about 29 years. We conclude that the spectra of 732 have
not changed within the observed time interval, which is consistent with our
results for 269 or 596.

##  6 203 Pompeja

For 203 Pompeja, a visible spectrum, visible spectrophotometric color, and NIR
spectrum were taken on 2021 January 29, Febrary 4, and January 14,
respectively (Hasegawa et al., 2021). Meanwhile, most of the data from
Sergeyev et al. (2022) for 203 were acquired from 2018 April 16 to May 15. The
visible spectra of 203, which were acquired by Sawyer (1991); Gaia
Collaboration et al. (in press)3, match the one of Sergeyev et al. (2022). The
slope of the spectrum of 203 obtained by Hasegawa et al. (2021) is redder than
that of a typical D type, but the slope of the spectrum obtained by Sergeyev
et al. (2022) is comparable to that of a T type (Figure 5). In addition, new
visible and NIR spectra of 203 were obtained on 2022 June 18, 19, 29, and 30,
and 2020 December 1 and 2022 June 10 and 27, respectively and classified as X
types. The above indicate that there is a large slope variation of the visible
and NIR spectra of 203 acquired in Hasegawa et al. (2021) with respect to the
other years. Moreover, the latitude and longitude of the Sub-Earth point at
the time of both observations by IRTF coincide (see Table 2). Table 2 was
based on the shape model of 203 from Appendix A.

Table 2: Sub-solar and observer coordinates of 203 Pompeja during photometric
and spectral observations

Observation time | Telescope | Sub-solar point | Sub-Earth point | Reference
---|---|---|---|---
|  |  ${\lambda_{1}}^{a}$ [°] |  ${\phi_{1}}^{a}$ [°] |  ${\lambda_{2}}^{b}$ [°] |  ${\phi_{2}}^{b}$ [°] |  ${\lambda_{1}}^{a}$ [°] |  ${\phi_{1}}^{a}$ [°] |  ${\lambda_{2}}^{b}$ [°] |  ${\phi_{2}}^{b}$ [°] |
1988/10/06 08:15 | McDonald | 60 |  $-$1 | 234 | 2 | 51 | 11 | 225 |  $-$10 | Sawyer (1991)
2018/04/16 14:37–54 | SkyMapper | 21–16 |  $-$4 | 202–197 | 4 | 15–10 |  $-$13 | 194–190 | 11 | Sergeyev et al. (2022)
2018/04/17 13:55 | SkyMapper | 32 |  $-$4 | 212 | 4 | 26 |  $-$13 | 205 | 11 | Sergeyev et al. (2022)
2018/04/21 14:14–30 | SkyMapper | 30–26 |  $-$5 | 211–207 | 4 | 25-21 |  $-$13 | 204-200 | 311 | Sergeyev et al. (2022)
2018/04/22 13:42 | SkyMapper | 39 |  $-$5 | 219 | 5 | 34 |  $-$13 | 213 | 11 | Sergeyev et al. (2022)
2018/05/15 14:11–12 | SkyMapper | 47 |  $-$9 | 228 | 8 | 49 |  $-$9 | 228-227 | 7 | Sergeyev et al. (2022)
2019/09/01 13:36 | SkyMapper | 334 |  $-$48 | 156 | 49 | 340 |  $-$50 | 162 | 50 | Sergeyev et al. (2022)
2019/09/07 09:48 | SkyMapper | 34 |  $-$48 | 216 | 48 | 44 |  $-$50 | 226 | 50 | Sergeyev et al. (2022)
2020/12/01 08:21 | IRTF | 332 | 29 | 153 |  $-$28 | 337 | 35 | 156 |  $-$32 | This study
2021/01/14 09:08 | IRTF | 357 | 36 | 179 |  $-$36 | 9 | 29 | 189 |  $-$26 | Hasegawa et al. (2021)
2021/01/29 12:55 | SAO(Spec) | 310 | 39 | 132 |  $-$38 | 326 | 28 | 146 |  $-$26 | Hasegawa et al. (2021)
2021/02/04 10:36 | SAO(Phot) | 348 | 39 | 170 |  $-$39 | 5 | 27 | 189 |  $-$26 | Hasegawa et al. (2021)
2022/06/10 05:41 | IRTF | 341 | 17 | 163 |  $-$17 | 357 | 31 | 175 |  $-$33 | This study
2022/06/18 23:27 | TRAPPIST | 80 | 16 | 262 |  $-$16 | 97 | 30 | 277 |  $-$32 | This study($BVR_{\rm C}I_{\rm C}$)
2022/06/19 01:02 | TRAPPIST | 57 | 16 | 239 |  $-$16 | 73 | 30 | 253 |  $-$32 | This study($BVR_{\rm C}I_{\rm C}$)
2022/06/19 02:43 | TRAPPIST | 32 | 16 | 213 |  $-$16 | 48 | 30 | 228 |  $-$32 | This study($BVR_{\rm C}I_{\rm C}$)
2022/06/27 08:08 | IRTF | 316 | 14 | 138 |  $-$15 | 332 | 29 | 152 |  $-$31 | This study
2022/06/29 00:18 | TRAPPIST | 75 | 14 | 257 |  $-$15 | 91 | 29 | 271 |  $-$31 | This study($BVR_{\rm C}I_{\rm C}z^{\prime}$)
2022/06/29 23:15 | TRAPPIST | 91 | 14 | 273 |  $-$14 | 107 | 28 | 287 |  $-$31 | This study($BVR_{\rm C}I_{\rm C}z^{\prime}$)
2022/06/30 00:57 | TRAPPIST | 66 | 14 | 248 |  $-$14 | 82 | 28 | 262 |  $-$30 | This study($BVR_{\rm C}I_{\rm C}z^{\prime}$)
2022/06/30 02:30 | TRAPPIST | 43 | 14 | 225 |  $-$14 | 58 | 28 | 239 |  $-$30 | This study($BVR_{\rm C}I_{\rm C}$)

Note. —

aPole solution 1: ${\lambda}_{1}$ = 315 [°], ${\beta}_{1}$ = $-$40 [°], Period
= 24.0550 [hr]

bPole solution 2: ${\lambda}_{2}$ = 132 [°], ${\beta}_{2}$ = $-$36 [°], Period
= 24.0551 [hr]

The fact that the NIR spectra of other asteroids obtained using the same solar
analogue stars on the same date as Hasegawa et al. (2021) are consistent with
previous spectra gives credit to this NIR spectrum of 203 (see Appendix B).
Additional visible data for three other asteroids obtained by the same surveys
as 203 are highly consistent with the data from Sergeyev et al. (2022) and
Gaia Collaboration et al. (in press) (see Appendix C). Therefore, it appears
that the visible spectrum and spectrophotometric color reported in Hasegawa et
al. (2021) are valid. Furthermore, the slope of the measured visible spectrum
(0.7–0.8 $\mu$m) is consistent with the slope in the NIR spectrum (0.8–0.9
$\mu$m), which further validates the observational result of Hasegawa et al.
(2021).

The first possible cause of the spectral differences is cometary activity, as
reported on 596. Therefore, the radial profiles of 203 around the observation
period of Hasegawa et al. (2021) were investigated by comparing the radial
profiles with field stars to detect a faint comet-like coma. However, no coma
is detected within the background detection limit of 25.1 mag arcsec-2 in the
$r$ band (see Appendix D). This evidence suggests that the spectral
differences may not be triggered by an unforeseen cometary activity.

The only difference between the IRTF observations is the latitude of the sub-
solar point. Sub-solar points were at relatively high latitudes444This is
applied to the first shape/pole model, whereas the second shape/pole solution
has the northern and southern hemispheres reversed. in 2021 observations
(Table 2). The existence of regions with a VRO-like spectrum in high latitudes
is necessary to explain the spectral differences. There may be a giant crater,
that is shaded under low sub-solar point and whose areas are not visible, such
as the giant crater on asteroid 253 Mathilde (Veverka et al., 1997), for
example. Since the VRO-like spectra are much brighter in the NIR than the
X-type spectra, even if the area of the crater illuminated by sunlight is
small, the feature may appear in the spectrum. An asteroid with surface
heterogeneity was also reported in Avdellidou et al. (2021). These authors
suggested that the very red spectrum may correspond to a fresh crater formed
by an impact as their studied asteroid 223 Rosa, also showed a very red
spectrum when observed at different epochs. Finding craters with a VROs-like
color on asteroids with an X-type surface implies that asteroids with a VROs
surface layer evolved to an X-type-like surface layer by space weathering. If
solar wind is the main cause of space weathering, the time of sunlight
exposure at the bottom of craters such as this one, where sunlight is less
directly illuminated, is much shorter than on other surfaces, so that the time
for sunlight penetration is relatively longer and the spectral features may
not disappear. Considering that VROs have the same spectral properties as RR
and IR class trans-Neptunian objects (Fulchignoni et al., 2008; Fraser et al.,
submitted), this supports the proposal of (Vernazza et al., 2021; Rivkin et
al., 2022) that D/T-type and dark X-complex asteroids originated from the same
transplanetary and/or transneptunian planetesimal region.

##  7 Conclusions

A comparative study of the spectra of the T/D-type asteroid 596 Scheila
before, immediately after, and 10 years after the 2010 impact event reveals
the following:

  * •

Fresh surfaces on dark asteroids with reddish spectra like 596 do not undergo
any noticeable spectral changes induced by space weathering over a 10-year
timescale.

  * •

This is the first time that the absence of spectral changes due to space
weathering has been measured over an accurate time interval for an actual
asteroid, as opposed to meteorite samples measured in laboratory simulations.

  * •

During the first decade of space weathering following a resurfacing event, the
spectral change progression of dark and reddish asteroids does not appear to
progress logarithmically but linearly.

Applying the findings of our study to VROs: 203 Pompeja, 269 Justitia, and 732
Tjilaki, we can suggest the following:

  * •

The spectrum of 203 Pompeja was found to be a VRO in the 2021 observations,
but an X type in the other observational epoch. The existence of a giant
crater with a VRO spectrum in one of the polar regions is suggested to explain
this spectral difference. This would imply that the spectra of VROs evolve
into dark X types as space weathering progresses over time.

  * •

Combining the results from the study of Fatka et al. (2022) on asteroid pairs
and our own results, it is possible that the spectral slope of VROs remains
mostly unchanged over a timescale of 300 years.

We would like to thank the referee for the careful review and constructive
suggestions, which helped us to improve the manuscript significantly. We are
grateful to Dr. Joshua P. Emery for sharing valuable 269 data and supporting
203 observation. We greatly appreciate Dr. Petr Fatka and Dr. Nicholas A.
Moskovitz for useful information about their observed NEAs. This work is based
on observations collected at the Infrared Telescope Facility, which is
operated by the University of Hawaii under contract 80HQTR19D0030 with the
National Aeronautics and Space Administration, Ishigakijima Astronomical
Observatory, National Astronomical Observatory of Japan, and at Seoul National
University Astronomical Observatory. The authors acknowledge the sacred nature
of Mauna kea and appreciate the opportunity to observe from the mountain. The
instrumentation at IAO was supported by a Grant-in-Aid for Scientific Research
on Priority Areas (19047003). TRAPPIST is a project funded by the Belgian
Fonds (National) de la Recherche Scientique (F.R.S.-FNRS) under grant PDR
T.0120.21. We thank the Las Cumbres Observatory and their staff for its
continuing support of the ASAS-SN project. ASAS-SN is supported by the Gordon
and Betty Moore Foundation through grant GBMF5490 to the Ohio State
University, and funded in part by the Alfred P. Sloan Foundation grant
G-2021-14192 and NSF grant AST-1908570. Development of ASAS-SN has been
supported by NSF grant AST-0908816, the Mt. Cuba Astronomical Foundation, the
Center for Cosmology and AstroParticle Physics at the Ohio State University,
the Chinese Academy of Sciences South America Center for Astronomy (CAS-SACA),
the Villum Foundation, and George Skestos. This work has made use of data from
the Asteroid Terrestrial-impact Last Alert System (ATLAS) project. ATLAS is
primarily funded to search for near earth asteroids through NASA grants
NN12AR55G, 80NSSC18K0284, and 80NSSC18K1575; byproducts of the NEO search
include images and catalogs from the survey area. The ATLAS science products
have been made possible through the contributions of the University of Hawaii
Institute for Astronomy, the Queen’s University Belfast, the Space Telescope
Science Institute, the South African Astronomical Observatory (SAAO), and the
Millennium Institute of Astrophysics (MAS), Chile. ZTF data were based on
observations obtained with the Samuel Oschin Telescope 48-inch and the 60-inch
Telescope at the Palomar Observatory as part of the Zwicky Transient Facility
project. ZTF is supported by the National Science Foundation under GrantNo.
AST-2034437 and a collaboration including Caltech, IPAC, the Weizmann
Institute for Science, the Oskar Klein Center atStockholm University, the
University of Maryland, Deutsches Elektronen-Synchrotron and Humboldt
University, the TANGOConsortium of Taiwan, the University of Wisconsin at
Milwaukee, Trinity College Dublin, Lawrence Livermore NationalLaboratories,
and IN2P3, France. Operations are conducted by COO, IPAC, and UW. This study
has utilized the SIMBAD database, operated at CDS, Strasbourg, France, and the
JPL HORIZONS ephemeris generator system, operated at JPL, Pasadena, USA. This
work is based on data provided by the Minor Planet Physical Properties
Catalogue (MP3C) of the Observatoire de la Côte d’Azur. F.D. and M.M. were
supported by the National Aeronautics and Space Administration under grant No.
80NSSC18K0849 and 80NSSC18K1004 issued through the Planetary Astronomy
Program. Any opinions, findings, and conclusions or recommendations expressed
in this letter are those of the authors and do not necessarily reflect the
views of the National Aeronautics and Space Administration. The work of J.H.
has been supported by the Czech Science Foundation through grant 20-08218S.
C.A. and M.D. acknowledge support from ANR ”ORIGINS” (ANR-18-CE31-0014). C.A.
and M.D. were Visiting Astronomers at the Infrared Telescope Facility, which
is operated by the University of Hawaii under contract 80HQTR19D0030 with the
National Aeronautics and Space Administration. E.J. is FNRS Senior Research
Associate. M.Im acknowledges the support from the Korea Astronomy and Space
Science Institute grant under the R&D program (Project No.2020-1-600-05)
supervised by the Ministry of Science and Technology and ICT (MSIT), and the
National Research Foundation of Korea (NRF) grant, No. 2020R1A2C3011091,
funded by MSIT. M.I. was supported by the NRF grant No. 2018R1D1A1A09084105.
This study was supported by JSPS KAKENHI (grant nos. JP20K04055, JP21H01140,
JP21H01148, and JP22H00179) and by the Hypervelocity Impact Facility (former
facility name: the Space Plasma Laboratory), ISAS, JAXA.

##  Appendix A Shape model of 203 Pompeja

In order to link the measured spectra of 203 Pompeja to their corresponding
regions on its surface (i.e., to derive the sub-observer points), the
knowledge of the rotation state (i.e., sidereal rotation period and
orientation of the spin axis) is necessary. However, 203’s shape model and
rotation state had not been derived so far. To fix this, we compiled available
optical data of 203 and applied the convex inversion. This light curve
inversion method developed by Kaasalainen & Torppa (2001); Kaasalainen et al.
(2001) was already successfully used for the 3D shape and rotation state
determination of more than 3,400 asteroids (see the Database of Asteroid
Models from Inversion Techniques,
DAMIT555https://astro.troja.mff.cuni.cz/projects/damit/, Durech et al. 2010).
Therefore, convex inversion is a well tested state-of-the art method for the
shape modeling of asteroids. Here we follow the standard modeling approach
described in details, for instance, in Hanuš et al. (2021); Athanasopoulos et
al. (in press).

Our optical dataset include 18 dense light curves from two apparitions: in
1983 (di Martino, 1984) and 2011/2012 (Pilcher et al., 2012). Especially the
latter dataset is highly valuable as the authors dedicated a large effort in
reliably deriving the rotation period of 203. The main difficulty stems from
the rotation period being very close to 24 hr, the rotation phase then cannot
be fully sampled from one location on the Earth, but rather multiple observers
at different longitudes have to participate in the campaign.

Furthermore, we supplemented the dense light curves with the sparse-in-time
photometric measurements from five different sky-surveys. In particular, we
used data from the Catalina Sky Survey (CSS; Larson et al., 2003), the US
Naval Observatory in Flagstaff (USNO-Flagstaff), the Asteroid Terrestrial-
impact Last Alert System (ATLAS; Tonry et al., 2018a, b; Heinze et al., 2018),
the All-Sky Automated Survey for Supernovae (ASAS-SN; Shappee et al., 2014;
Kochanek et al., 2017; Hanuš et al., 2021), and the Gaia Data Release 2 (Gaia
DR2; Gaia Collaboration et al., 2018).

The optical dataset for 203 allowed us to derive its unique spin state and 3D
shape model solution. 203 rotates with a sidereal rotation period of (24.05502
$\pm$ 0.00005) hr and we provide two symmetric pole solutions following the
standard ambiguity of the inversion method (Kaasalainen, 2004). The appearance
of the two shape models in shown in Fig. 6.

Figure 6: Convex shape models of 203 Pompeja. While the left panel represents
the pole solution ${\lambda}_{1}$ = 315 [°], ${\beta}_{1}$ = $-$40 [°], the
right panel shows the pole solution ${\lambda}_{2}$ = 132 [°], ${\beta}_{2}$ =
$-$36 [°]. The left and center views are equatorial with a 90∘ rotation, while
the right view is pole-on. Both shape models appear very similar.

##  Appendix B A NIR observation of 203 Pompeja in 2021 January

The spectrum of 203 Pompeja measured on 2021 January 14 (Hasegawa et al.,
2021) is much redder than usual asteroids, which may cast doubt on the
validity of the observations. One possibility could be that the spectrum was
red due to slit misalignment. However, if the asteroid is out of alignment,
the spectrum is bluing instead of reddening (Marsset et al., 2020). Therefore,
in order for the asteroid to become red, it is the solar analogue star
observations that needed to be misaligned. On the night we observed 203 (2021
January 14), we also observed five solar analogue stars for the spectral
division, four of which were are spectrally very consistent (the fifth star
was discarded). In addition, we observed NEAs 99942 Apophis and 2020
$\mathrm{WU_{5}}$ on the same night as 203 and their respective spectra were
consistent with previous and more recent observations (Figure 7). This provide
strong credit to our 2021 observations of 203. Future spectroscopic
observations and high-angular resolution imaging of 203 acquired under high
sub-observer latitudes will help confirming or dismissing the existence of a
fresh polar crater with a VRO spectrum on 203.

Figure 7:  Spectra of 99942 Apophis (left panel) and 2020 $\mathrm{WU_{5}}$
(right panel). Data of 203 Pompeja, 99942 Apophis, and 2020 $\mathrm{WU_{5}}$
taken on 2021 January 14. The spectra of 99942 and 2020 $\mathrm{WU_{5}}$ were
measured four and one times, respectively (Binzel et al., 2009, 2019; Marsset
et al., 2022), in addition to their measurement on the same day as 203
(Hasegawa et al., 2021). The five spectra of 99942 and the two spectra of 2020
$\mathrm{WU_{5}}$ are found to be in agreement. Square brackets in these
panels indicate the observation dates. The black arrows indicate the slope
uncertainty from Marsset et al. (2020) centered at the reflectance value of
the spectra at 2.40 $\mu$m, as shown by the black dashed lines.

##  Appendix C Visible observations of three asteroids after 2020

Spectroscopic and spectrophotometric observations of three MBAs: 212 Medea,
328 Gudrun, and 769 Tatjana were performed with the Shelyak PF0021VIS–LISA
spectrometer and the SBIG STX-16803 camera equipped with $V$-, $R_{\rm C}$-,
and $I_{\rm C}$-band filters on the 1.0 m telescope at Seoul National
University Astronomical Observatory (SAO), Seoul, Republic of Korea (Im et
al., 2021) in 2020 or later. The procedures for the visible observations and
reduction are as described in Hasegawa et al. (2021).

The spectroscopic data of this study yield classification of 212 Medea, 328
Gudrun, and 769 Tatjana as X, Cgh, and Xc types, respectively. The acquired
spectra match previously obtained spectra (Figure 8). Therefore, it appears
that the visible spectrum and spectrophotometric color reported in Hasegawa et
al. (2021) are valid.

Figure 8: Spectra of 212 Medea, 328 Gudrun, and 769 Tatjana. Spectra of these
asteroids were obtained at SAO in 2020–2021. Previous spectra were derived
from Sergeyev & Carry (2021); Sergeyev et al. (2022); Gaia Collaboration et
al. (in press).

##  Appendix D Search for a Coma of 203 Pompeja

To confirm the comet-like activity of 203 Pompeja, radial brightness profiles
of 203 and field stars around the observation date of Hasegawa et al. (2021)
were examined. For comparison, images in the $r$ band acquired at the 48-inch
(1.2 m) Samuel Oschin Schmidt telescope in the Zwicky Transient Facility
(Masci et al., 2019) were
utilized666https://www.ipac.caltech.edu/doi/irsa/10.26131/IRSA539. The radial
brightness profiles of 203 and field stars are consistent over arange of about
three orders of magnitude, and no comet-like coma was detected around the
observation date of Hasegawa et al. (2021) (Figure 9).

Figure 9:  Radial brightness profiles of 203 Pompeja obtained by the Zwicky
Transient Facility on 2020 December 10 (top left), 17 (top middle), 22 (top
right) and 2021 January 5 (bottom left), 13 (bottom middle), and 19 (bottom
right). These profiles are normalized at 2.5” from the photo center and
compared with field stars. The FWHM of the images used was roughly 2”–3”.
Since 203 was almost saturated, brightness profiles within 2.5” from the photo
center are not given.

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