# Heliocentric Distance Dependence of Zodiacal Light Observed by Hayabusa2#

K.Ts.Kohji Tsumura  S.M.Shuji Matsuura  K.Sa.Kei Sano  T.I.Takahiro Iwata
H.Y.Hajime Yano  K.K.Kohei Kitazato  K.Ta.Kohji Takimoto  M.Y.Manabu Yamada
T.M.Tomokatsu Morota  T.K.Toru Kouyama  M.H.Masahiko Hayakawa  Y.Y.Yasuhiro
Yokota  E.T.Eri Tatsumi  M.M.Moe Matsuoka  N.S.Naoya Sakatani  R.H.Rie Honda
S.K.Shingo Kameda  H.Su.Hidehiko Suzuki  Y.C.Yuichiro Cho  K.Y.Kazuo Yoshioka
K.O.Kazunori Ogawa  K.Sh.Kei Shirai  H.Sa.Hirotaka Sawada  S.S.Seiji Sugita
Department of Natural Sciences, Faculty of Science and Engineering, Tokyo City
University, 158-8557 Tokyo, Japan  Department of Physics, School of Science
and Technology, Kwansei Gakuin University, 669-1337 Hyogo, Japan  Department
of Space Systems Engineering, School of Engineering, Kyushu Institute of
Technology, 804-8550 Fukuoka, Japan  Institute of Space and Astronautical
Science, Japan Aerospace Exploration Agency, 252-5210 Kanagawa, Japan
Department of Space and Astronautical Science, The Graduate University of
Advanced Studies, 240-0193 Kanagawa, Japan  Department of Computer Science and
Engineering, The University of Aizu, 965-8580 Fukushima, Japan  Planetary
Exploration Research Center, Chiba Institute of Technology, 275-0016 Chiba,
Japan  Department of Earth and Planetary Science, University of Tokyo,
113-0033 Tokyo, Japan  Digital Architecture Research Center, National
Institute of Advanced Industrial Science and Technology, 135-0064 Tokyo, Japan
nstituto de Astrofisica de Canarias, University of La Laguna, Tenerife, Spain
Geological Survey of Japan, National Institute of Advanced Industrial Science
and Technology, 305-8567 Ibaraki, Japan  Center for Data Science, Ehime
University, 790-8577 Ehime, Japan  Rikkyo University, 171-8501 Tokyo, Japan
Meiji University, 214-8571 Kanagawa, Japan  Department of Complexity Science
and Engineering, University of Tokyo, 277-8561 Chiba, Japan  JAXA Space
Exploration Center, Japan Aerospace Exploration Agency, 252-5210 Kanagawa,
Japan  Kobe University, 657-8501 Hyogo, Japan

###### Abstract

Zodiacal light (ZL) is sunlight scattered by interplanetary dust particles
(IDPs) at optical wavelengths. The spatial distribution of IDPs in the Solar
System may hold an important key to understanding the evolution of the Solar
System and material transportation within it. The number density of IDPs can
be expressed as $n(r)\sim r^{-\alpha}$, and the exponent $\alpha\sim 1.3$ was
obtained by previous observations from interplanetary space by Helios 1/2 and
Pioneer 10/11 in the 1970s and 1980s. However, no direct measurements of
$\alpha$ based on ZL observations from interplanetary space outside Earth’s
orbit have been performed since then. Here, we introduce initial results for
the radial profile of the ZL at optical wavelengths observed over the range
0.76-1.06 au by ONC-T aboard the Hayabusa2# mission in 2021-2022. The ZL
brightness we obtained is well reproduced by a model brightness, although
there is a small excess of the observed ZL brightness over the model
brightness at around 0.9 au. The radial power-law index we obtained is
$\alpha=1.30\pm 0.08$, which is consistent with previous results based on ZL
observations. The dominant source of uncertainty arises from the uncertainty
in estimating the diffuse Galactic light (DGL).

zodiacal light,

interplanetary dust,

Hayabusa2#,

###### keywords:

Full Paper

Graphical Abstract

##  1 Introduction

The zodiacal light (ZL) is sunlight scattered by interplanetary dust particles
(IDPs) at optical wavelengths, and it is a major constituent of the diffuse
celestial brightness. A continuous supply of IDPs is necessary to sustain the
diffuse brightness because IDP is removed from the Solar System due to the
Poynting-Robertson (PR) effect and by radiation pressure from the Sun [1, 2].
Possible sources for this supply are asteroid collisions [3, 4, 5] or cometary
ejections [6, 7, 8], but the relative ratios of the contributions from these
sources are still unknown. Dust of interstellar origin also contributes
$\sim$10% to the total amount of IDP [9]. Thus, observational constraints that
can tell the differences among these sources are important for a better
understanding the origin and characteristics of IDPs and of the way planetary
and exoplanetary systems evolve with time [10, 11].

Historically, extensive ZL observations were conducted from ground-based
telescopes at high-altitude sites in the 1960s and 1970s [12, 13], but the
accuracy of these ZL observations is limited due to atmospheric emission. In
contrast, space-based platforms eliminate atmospheric contamination and
provide precise ZL measurements [14, 15, 16, 5, 17, 18, 19, 20, 21]. The ZL is
the only sky-brightness component that is not fixed on the celestial sphere.
In general, the ZL is smoothly distributed, and its small-scale spatial
structures are only at the level of a few percent owing to the smooth spatial
distribution of IDP as a smooth cloud [22]. The plane of symmetry of the
smooth cloud is slightly inclined to the ecliptic plane because of the Jovian
orbit. Seasonal variations in ZL occur for an Earth-based observer due to the
orbital motion of the Earth, which changes the heliocentric distance and the
position of the observer with respect to the symmetry plane. A detailed IDP
distribution model has been established based on the seasonal variation of the
ZL [23, 24].

The number density ($n$) of IDP is presumed to be of a form that is separable
into radial and vertical terms;

$n(r,\beta)=n_{0}\left(\frac{r}{r_{0}}\right)^{-\alpha}f(\beta),$ (1)

where $n_{0}$ is the reference number density of IDPs in the symmetry plane at
the heliocentric distance $r_{0}$, and $f(\beta)$ denotes the vertical
distribution as a function of an elevation angle $\beta$ from the symmetry
plane [25]. The assumption that the vertical distribution of the IDPs depends
only on $\beta$ is suggested by the fact that the PR effect does not affect
the orbital inclinations of particles as they spiral into the Sun. The radial
power-law is induced by the radial distribution expected for particles under
the influence of the PR effect, which results in $\alpha=1$ for dust bound in
a circular orbit [2]. When dust-grain sizes are reduced by sublimation near
the Sun, such smaller dust particles are expelled from the Solar System as
$\beta$-meteoroids by radiation pressure [26, 27, 28]. The radial profile of
$\beta$-meteoroids is expected to follow a power law with $\alpha=2$ [29]. The
relative ratio of these two components remains an open issue and may hold an
important key for understanding the evolution of the IDP distribution [30,
31].

The ZL brightness $I_{ZL}$ can be modeled as the integral of scattered
sunlight along the line of sight:

$I_{ZL}=\int F_{\odot}(r)n(r)A\Phi(\theta)dl,$ (2)

where $F_{\odot}(r)\sim r^{-2}$ is the Solar flux at the distance $r$ from the
Sun, $A$ is the albedo of the IDP, $\Phi(\theta)$ is the phase function at the
scattering angle $\theta$, and $dl$ is an increment along the line of sight.
If the scattering properties (size and albedo) of IDPs do not change
significantly with heliocentric distance, the heliocentric dependence of ZL
toward the antisolar direction on the symmetry plane can be written as
$I_{ZL}\sim r^{-(\alpha+1)}$. If the line of sight is not oriented in the
antisolar direction, the heliocentric dependence of $I_{ZL}$ becomes much more
complex, since it depends on the phase function $\Phi(\theta)$ for which
$\theta$ will vary. In addition, a heliocentric dependence of the local albedo
of the IDPs has also been reported [32], which makes the heliocentric
dependence of $I_{ZL}$ even more complex.

Direct observations of the radial power-law index $\alpha$ based on ZL
observations were performed from spacecraft outside Earth’s orbit in the 1970s
and 1980s. Pioneer 10/11 observations of the ZL at 1-3.3 au gave
$\alpha=$1-1.5. More specifically, a single power-law model with $\alpha\sim
1$ and a cutoff near 3.3 au gives the best fit to the observational data,
although a two-component model with $\alpha\sim 1.5$ and increased IDP in the
asteroid belt fits the data equally well [33]. Helios 1/2 observations of the
ZL at 0.3-1 au gave $\alpha=1.3\pm 0.05$, although $\alpha=1.35$ gives a
better fit for small solar elongations ($<50^{\circ}$), and $\alpha=1.25$ is
more appropriate for large solar elongations ($>100^{\circ}$) [34, 35]. ZL
observations from spacecraft outside Earth’s orbit have not been performed
following these missions. The Japanese Venus orbiter Akatsuki tried but could
not detect the ZL due to insufficient cooling of the sensor [36].

Some IDP distribution models were developed based on observations of the all-
sky ZL brightness and its seasonal variation from geocentric orbit. In
particular, observations from the Cosmic Background Explorer (COBE) yielded
$\alpha=1.34\pm 0.022$ [23] and $\alpha=1.22$ [24], and observations by AKARI
gave $\alpha=1.59\pm 0.02$ [37]. These observations of the ZL were performed
at 1 au, so the accuracy in determining $\alpha$ was worse than that obtained
by direct observations from interplanetary space.

The value of $\alpha$ has also been determined based on the observations of
the inner ZL or F-corona. Observations of the inner ZL by Clementine from
lunar orbit while the Sun was in eclipse behind the Moon yielded
$\alpha=1.45\pm 0.05$ [38]. Values of $\alpha$ from 1.31 to 1.35 were obtained
from F-corona observations at elongations ranging from 0.07 to 0.45 au from
the Sun [39] by the Heliospheric Imager-1 [40] onboard the Solar TErrestrial
RElations Observatory-A (STEREO-A) orbiting the Sun at approximately 1 au. In
addition, $\alpha=1.31$ was obtained by F-corona observations between 0.1 and
0.4 au [41] by the Widefield Imager for Solar Probe inner telescope (WISPER-1)
[42] onboard the Parker Solar Probe (PSP) when it passed perihelion at
0.16-0.25 au. These results are limited to dust distributions close to the
Sun.

A technique for studying the distribution and properties of IDPs independent
of the ZL observation is in-situ dust counting using dedicated dust detectors.
The size distribution of IDPs was studied by the in-situ dust-counting method
and it was suggested that large (10-100 $\mu$m) dust is dominant around 1 au
[43, 44]. The ZL brightness is indicative of the IDP distribution in the inner
Solar System, where the IDP density is substantial, and the IDP distribution
derived from these ZL observations is confined to the inner Solar System ($<5$
au). Conversely, dust distribution in the outer Solar System has been
investigated by the in-situ dust-counting method [45, 46].

This paper introduces the IDP distribution based on the ZL observations from
the Hayabusa2# mission at 0.76-1.06 au performed in 2021-2022. These are the
first successful observations of the ZL from outside Earth’s orbit in the last
40 years.

Figure 1: A comparison of the relative transmissivity. The solid line shows
the bandpass of ONC-T/Hayabusa2# wide-band [47], the dashed line shows the
bandpass of Gaia G-band [48], and the dotted line shows the bandpass of
LORRI/New Horizons [49].

##  2 Data Acquisition and Reduction

###  2.1 Hayabusa2# overview

Hayabusa2 is the second Japanese asteroid-sample-return mission. The Hayabusa2
spacecraft was launched in December 2014 and successfully arrived at asteroid
(162173) Ryugu in June 2018. After extensive scientific observations for
$\sim$1.5 years, it departed from Ryugu in November 2019 and successfully
brought the capsule containing Ryugu samples back to Earth in December 2020
[50, 51]. With the successful main mission of the sample return completed, an
extended mission named Hayabusa2# (SHARP; Small Hazardous Asteroid
Reconnaissance Probe) was initiated to explore new asteroids; it will perform
a fly-by of (98943) 2001 CC21 in July 2026 and a rendezvous with 1998 KY26 in
July 2031 [52]. Some scientific observations including ZL observations will be
performed during this long cruising phase [53].

The Optical Navigation Camera (ONC) onboard Hayabusa2 consists of one
telescopic camera (ONC-T) and two wide-angle view cameras (ONC-W1/W2) [54, 55,
47, 56, 57], and it was used for both global and local high-resolution optical
observations of Ryugu [58]. The ONC was carefully calibrated both before and
after launch, and it remains in good condition after contact with the surface
of Ryugu during the two touchdowns for sampling. In this study, we used ONC-T
for the ZL observations. The field of view of ONC-T is $6.27\times 6.27$ deg2,
which is covered with a $1024\times 1024$ pixel region of a CCD detector [54].
The longest exposure time of ONC-T was 178 sec, which we used for the ZL
observations in this study. ONC-T has a wheel system that rotates seven color-
bandpass filters and one wide clear filter (a panchromatic glass window). We
used the wide-band filter ($\lambda$ = 612 nm and $\Delta\lambda$ = 448 nm,
see Figure 1) for the ZL observations, with a v-band filter ($\lambda$ = 550
nm and $\Delta\lambda$ = 28 nm) for stray light subtraction (see Section 2.5).

Figure 2: Hayabusa2# orbit in the J2000EC inertial frame before the Earth
swing-by on December 2027. The regions suitable for ZL observations are shown
in red, and unsuitable regions owing to pointing toward the Galactic plane
(Galactic latitude $<20^{\circ}$) and the Galactic center (Galactic longitude
$<20^{\circ}$) are shown in blue and green, respectively. Figure 3: (a)
Heliocentric distance of the Hayabusa2 spacecraft and (b) its ecliptic
longitude, (c) Galactic longitude, and (d) Galactic latitude of the antisolar
direction during the period from April 2021 to March 2023. The regions
suitable for ZL observations are shown in red, and unsuitable periods owing to
pointing toward the Galactic plane (Galactic latitude $<20^{\circ}$) and the
Galactic center (Galactic longitude $<20^{\circ}$) are shown in blue and
green, respectively.
Table 1: Spacecraft positions and observed fields

| Position of Spacecraft [au] | Observed Field [deg]
---|---|---
Date | X | Y | Z | R(1) | (RA, Dec) | (Elon, Elat)(2) | (Glon, Glat)(3) | Solar Elongation
2021-08-23 | 1.023 | 0.252 | -0.069 | 1.053 | ( 15.93, -3.65) | ( 13.25, -9.64) | (130.58, -66.34) | 174.10
2021-09-20 | 0.821 | 0.633 | -0.047 | 1.037 | ( 37.06, 5.46) | ( 36.52, -8.71) | (162.23, -49.88) | 173.81
2021-11-29 | -0.261 | 0.846 | 0.035 | 0.886 | (104.42, 18.72) | (103.67, -4.03) | (196.79, 9.75) | 172.78
2021-12-06 | -0.375 | 0.781 | 0.042 | 0.866 | (112.96, 18.16) | (111.80, -3.56) | (200.79, 16.88) | 172.57
2021-12-28 | -0.664 | 0.468 | 0.057 | 0.812 | (142.74, 12.54) | (141.02, -2.05) | (219.94, 41.00) | 172.84
2022-01-24(4) | -0.774 | -0.067 | 0.055 | 0.777 | (198.38, 3.56) | (195.57, 10.49) | (316.49, 65.85) | 167.64
2022-02-14(4) | -0.625 | -0.474 | 0.037 | 0.784 | (204.83, -14.26) | (208.19, -3.68) | (320.08, 47.01) | 169.01
2022-04-18 | 0.451 | -0.810 | -0.048 | 0.927 | (297.25, -31.48) | (293.38, -10.22) | ( 8.92, -25.26) | 170.79
2022-05-16 | 0.834 | -0.535 | -0.071 | 0.991 | (327.89, -25.02) | (321.52, -11.33) | ( 25.45, -49.77) | 170.76
2022-06-20 | 1.039 | -0.024 | -0.076 | 1.039 | (357.99, -13.31) | (352.80, -11.40) | ( 74.84, -70.40) | 170.74
2022-07-04 | 1.028 | 0.193 | -0.071 | 1.046 | ( 8.76, -8.29) | ( 4.74, -11.08) | (110.55, -70.76) | 170.74
2022-08-01 | 0.854 | 0.587 | -0.051 | 1.036 | ( 30.13, 1.62) | ( 28.62, -10.00) | (155.74, -56.78) | 170.76
2022-08-29 | 0.505 | 0.858 | -0.021 | 0.996 | ( 53.46, 10.57) | ( 53.73, -8.38) | (174.72, -35.49) | 170.80
2022-10-17 | -0.318 | 0.811 | 0.038 | 0.871 | (106.56, 17.81) | (105.80, -4.72) | (198.51, 11.20) | 170.88
2022-11-14 | -0.683 | 0.415 | 0.057 | 0.799 | (144.53, 10.87) | (143.23, -3.08) | (222.98, 41.84) | 170.95
2022-12-12 | -0.750 | -0.154 | 0.052 | 0.766 | (184.25, -5.34) | (186.02, -3.21) | (287.28, 56.48) | 171.00

  * 1

(1) $R=\sqrt{X^{2}+Y^{2}}$.

  * 2

(2) Ecliptic coordinate.

  * 3

(3) Galactic coordinate.

  * 4

(4) These data were excluded from the analysis owing to the presence of
additional stray light (see Section 2.5).

###  2.2 Observation Fields

The Hayabusa2 spacecraft followed an elliptical orbit over the range 0.76-1.06
au before the Earth swing-by on December 2027, as shown in Figure 2 and 3
[52]. The spacecraft maintained an attitude in which the solar-array paddle
(+Z direction) was pointed toward the Sun during this period, and we performed
the ZL observations during periods when the ion engines were not in operation.
Since ONC-T points toward the -Z direction, the ZL is observed toward the
antisolar direction. This is an advantage of our ZL observations over past
observations because previous ZL observations in interplanetary space were
made at various solar elongation angles, making it difficult to distinguish
whether the ZL changes were due to changes in the heliocentric distance or in
the solar elongation. In our ZL observations, the change in ZL brightness due
to the solar elongation was minimized by observing the ZL at a nearly constant
solar elongation (see Table 1).

Stray light is produced when sunlight hits the radiator that cools the ONC-T
detector from a certain range of directions [55, 47]. Thus, the ZL
observations need to be conducted in a “stray-light-avoidance attitude”, in
which the -X side and +Y side of the spacecraft are illuminated by the Sun.
For this reason, the actual directions of our ZL observations are shifted from
the antisolar direction by $\sim$10 degrees. Table 1 summarizes the observed
fields and the position of the spacecraft when the observations were
conducted.

Periods when the observable direction (antisolar direction) is pointed toward
either the Galactic plane or the Galactic center are not suitable for ZL
observations because the Galactic brightness is too strong (see Section 3.1
and 3.2). Thus, we define as unsuitable ZL observation periods those in which
the antisolar direction is pointing toward (1) Galactic latitude $<20^{\circ}$
(the blue zones in Figures 2 and 3) or (2) Galactic longitude $<20^{\circ}$
(the green zones in Figures 2 and 3). The ZL observations were made
approximately once a month during the time suitable for ZL observations (the
red zones in Figures 2 and 3).

###  2.3 Acquired Images

The ONC-T detector has a $1024\times 1024$ pixel imaging region, with a
$16\times 1024$ pixel masked regions termed “optical black” on each side as a
dark reference [54]. The two optical-black images are combined and treated as
one $32\times 1024$ pixel optical-black image. Raw images acquired by ONC-T
are processed in a sequence of steps to calibrate the image data. In this
work, we used L2a-level images, which are raw FITS images with header
information containing the spacecraft system housekeeping data and ONC status
data (the temperatures of the detector, lens system, and electronics as well
as the voltages of the electronics, etc.) [47]. The signal from each pixel is
provided in 16-bit digital numbers (DN).

One ZL observation dataset includes one bias image $B(x,y)$, three wide-band
images $W_{i}(x,y)$ ($i=$1-3), two v-band images $V_{j}(x,y)$ ($j=$1-2), and
their respective optical-black images ($Bb(x,y)$, $Wb_{i}(x,y)$, and
$Vb_{j}(x,y)$), as shown in Figure 4. For the ZL observations on 2021-11-29,
2021-12-06, 2021-12-28, 2022-01-24, and 2022-02-14, as part of the calibration
operations we acquired two data sets to monitor the stability of the ONC-T
sensitivity after it was turned on.

Figure 4: Dataset acquired for one ZL observation. One dataset includes one
bias image $B(x,y)$, three wide-band images $W_{i}(x,y)$ ($i=$1-3), two v-band
images $V_{j}(x,y)$ ($j=$1-2), and their respective optical-black images
($Bb(x,y)$, $Wb_{i}(x,y)$, and $Vb_{j}(x,y)$).

###  2.4 Dark-Current Subtraction

Dark-current subtraction is essential for the measurement of diffuse radiation
such as the ZL. We estimated the dark current for each image in our dataset
from the corresponding dark image, which we obtained from the optical-black
image by subtracting a bias image ($Wb_{i}(x,y)-Bb(x,y)$ and
$Vb_{j}(x,y)-Bb(x,y)$). We then created a histogram of the dark image, and we
take its peak position to be the value of the dark current for that image. We
fitted the histogram of the dark image with a Gaussian function. The peak
position of the histogram corresponds to the mode of the dark image. Using
this Gaussian-fitting procedure, we eliminated bad pixels such as those due to
leakage of light from the imaging region or due to hot pixels caused by
cosmic-ray hits. We expressed the resulting dark-current values for the wide-
band and v-band images as $I_{\textrm{dark}}^{W_{i}}$ and
$I_{\textrm{dark}}^{V_{j}}$, respectively. As an example, we found the dark
current of the first bias-subtracted optical-black image taken on 2022-08-29
to be $I_{\textrm{dark}}^{W_{1}}=5.92$ DN from the peak position of the
histogram, as shown in Figure 5.

Next, the obtained dark current and the bias image are subtracted from the
imaging region, yielding three wide-band subtracted images and two v-band
subtracted images in one data set. From the wide-band images, we generated a
single reduced image $W(x,y$) from three subtracted images by taking the
median of each pixel:

$W(x,y)=\textrm{median}[W_{i}(x,y)-(B(x,y)+I_{\textrm{dark}}^{W_{i}})].$ (3)

This median procedure removes many hot pixels caused by cosmic-ray hits. Since
there are only two v-band images, we cannot use the median procedure for them.
Instead, we generated a single reduced image $V(x,y)$ by taking the minimum of
each pixel to reduce hot pixels caused by cosmic-ray hits:

$V(x,y)=\textrm{min}[V_{j}(x,y)-(B(x,y)+I_{\textrm{dark}}^{V_{j}})].$ (4)

Figure 5: Dark-current estimation. A histogram of the first bias-subtracted
optical-black image, $Wb_{1}(x,y)-Bb(x,y)$, obtained on 2022-08-29 (black) and
its best-fit Gaussian function (red). We obtained the dark current as
$I_{\textrm{dark}^{W_{1}}}=5.92$ DN from the peak position of this histogram.

###  2.5 Stray-Light subtraction

Since stray light occurs in ONC-T images when the spacecraft is at certain
attitudes [55, 47], we performed all ZL observations in stray-light-avoidance
attitudes (Section 2.2). However, weak stray light remains even in this case.
Figures 6a and 6b the reduced wide-band image $W(x,y)$ and v-band image
$V(x,y)$ obtained on 2021-08-23, respectively, and the stray-light patterns
can be seen clearly in these images. It is known that the intensity and
pattern of the stray light do not depend on the filter selection [55]. Thus,
we subtracted the v-band image $V(x,y)$ as a stray-light reference frame from
the wide-band image $W(x,y)$ to remove the remaining stray light, as shown in
Figures 6c and 6d. Since the ZL signal in the v-band image is estimated to be
less than 1 DN, there is little impact on the scientific analysis of the ZL
due to this stray-light-removal procedure.

We found an additional stray-light pattern in the data obtained on 2022-01-24
and 2022-02-14, as shown in Figure 7. Because this stray light appears only in
the wide-band image, it cannot be removed by the v-band subtraction procedure.
The source of this additional stray light is thought to be light scattered at
the inner wall of the entrance hole of the ONC-T hood, as is indicated by the
shape of the stray light (circular pattern). For this reason, we excluded data
from these two days from subsequent analyses.

Figure 6: Data-reduction procedure. (a): The wide-band image generated by
taking the median of the three dark-subtracted wide-band images. (b): The
v-band image generated by taking the minimum of two dark-subtracted v-band
images as a reference frame for the stray light. (c): The stray-light-
subtracted image $W(x,y)-V(x,y)$. (d): Same as (c). (e): A normalized flat-
field image [55]. (f): The flat-field-corrected image $SKY(x,y)$, which is the
final reduced image of the sky used for scientific analysis. Figure 7: The
additional stray light pattern found in the data obtained on 2022-01-24 and
2022-02-14.

###  2.6 Flat-Field Correction

The stray-light-subtracted image $W(x,y)-V(x,y)$ clearly shows a limb-
darkening pattern (Figures 6c and 6d), which is the same pattern as in the
flat-field image (Figure 6e) [47, 59]. This fact means that the detector is
uniformly illuminated from the front of the optics, showing that the sky
brightness has certainly been detected by the ONC-T. We corrected this limb-
darkening pattern by dividing the image by the normalized flat-field image
$FLAT(x,y)$, as shown in Figure 6f. Since we have not created a wide-band
flat-field image, we used the v-band flat-field image instead. The wavelength
dependence of the flat-field image is negligible because the detector is
identical for both bands. The image obtained after the flat-field correction
$SKY(x,y)$ is the final reduced image of the sky used for scientific analysis:

$SKY(x,y)=\frac{W(x,y)-V(x,y)}{FLAT(x,y)}.$ (5)

###  2.7 Sensitivity Calibration Using Stars

Degradation of the ONC-T sensitivity was reported after the two touchdown
operations on the asteroid Ryugu [56, 57]. Consequently, we monitored and
calibrated the sensitivity of the ONC-T in our data using the field stars in
our images. For this sensitivity calibration, we used the $W(x,y)/FLAT(x,y)$
image and not the $SKY(x,y)$ image. This is because the flux for the bright
stars used in this calibration process in case of the $SKY(x,y)$ image is
unsuitable for the sensitivity calibration as the v-band signal of the bright
stars has been subtracted in the $SKY(x,y)$ image. Although the
$W(x,y)/FLAT(x,y)$ image includes the stray light described in Section 2.5, it
can be removed by the aperture-photometry procedure described below.

The sensitivity-calibration procedure for the wide-band is as follows. First,
we solved the astrometry of the images from the distribution of the stars
using the astrometry-calculation code Astrometry.net [60]. Next, we matched
the bright stars in the image with those in the Gaia Data Release 3 (DR3)
catalog [61, 62]. This catalog is suitable for our dataset because the
wavelength coverage of Gaia’s G-band, which is an unfiltered, white-light
photometric band, is similar to that of our wide-band filter on ONC-T (Figure
1). The Gaia DR3 catalog contains around $1.806\times 10^{9}$ sources, with a
limiting magnitude of about $G\sim 21$ mag, with uncertainties of $\sim$0.3
mmag for $G<13$ mag, 1 mmag at $G=17$ mag, and 6 mmag at $G=20$ mag. We
selected stars that meet the following criteria:

  1. (a)

The selected stars are in regions with stray-light intensities less than 20 DN
in the $V(x,y)$ image,

  2. (b)

The selected stars are in the region with normalized flat-field values greater
than 0.8, and

  3. (c)

The selected stars have fluxes between the 6th and 9th AB magnitude in the
G-band in the Gaia DR3 catalog.

Criteria (a) and (b) reduce the uncertainty caused by the reduction processes
of stary-light subtraction and flat-field correction. The fraction of the area
satisfying both criteria (a) and (b) is approximately 44% of the total
detector area in the central region of the detector. Criterion (c) reduces the
uncertainty in the photometry by selecting stars that have sufficient signal
but are not saturated. The ONC-T detector is known to be linear up to
$\sim$3000 DN, with $<1$% deviation [47], and the signal value of even the
brightest pixel in an image of a 6th-magnitude star is approximately within
this range.

Figure 8: Aperture photometry of a star, as used for sensitivity calibration.
A boxed region of $41\times 41$ pixels centered on the selected star from the
catalog is cut out from the $W(x,y)/FLAT(x,y)$ image (left), and it is then
divided into a region containing the central star (center), which is used for
the aperture photometry of the star, and the surrounding background region
(right), which is used to estimate the background brightness and its noise.

For aperture photometry, a boxed region of $41\times 41$ pixels centered on
the selected star is cut out, and this boxed region is divided into a region
centered on the star and the surrounding background region, as shown in Figure
8. The radius of the circle used to cut out the star at the center is adjusted
according to the G-band brightness of this star in the catalog. Next, all
pixels greater than 20 DN are masked, as they are considered to be other
astronomical objects (stars and galaxies) or hot pixels caused by cosmic-ray
hits. We applied additional masks using a $\sigma$-clipping procedure to
remove the remaining bright pixels. Then we examined all the masked images by
eye and masked any additional remaining bad pixels. Subsequently, we
calculated the background brightness and its noise by computing the average
and standard deviation of the masked background region, and we subtracted the
background brightness from the star region. We then calculated the flux from
the central star by calculating the sum of the masked and background-
subtracted star region and estimated its uncertainty based on the background
noise. Figure 9 shows the relation between the G-band fluxes of the selected
stars from the catalog and their detected signal in the 2022-12-12 data. A
good linear relation between them exists in all the observed data, and we
obtained the sensitivity in units of (DN/sec)/(W/m2/$\mu$m/sr) by taking their
ratio.

Figure 9: The sensitivity calibration. The linear relationship between the
G-band flux of the selected stars and their detected signals in the
$W(x,y)/FLAT(x,y)$ image of the 2022-12-12 wide-band data as an example of the
sensitivity calibration. The red line shows the best fit to the data, and the
dashed lines show $\pm 1\sigma$ deviations. Figure 10: Sensitivity variation
of the wide-band (left) and v-band (right) images. The red solid lines show
the average of these sensitivities and the red dashed lines show their
$1\sigma$ standard deviations. The black dashed lines show the sensitivity
after the return to Earth [57].

Figure 10 (left) shows the sensitivity calculated from our observed data as a
function of time. The sensitivity obtained after the return to the Earth [57]
is also shown. This figure shows that the degradation of the sensitivity has
stopped and that the sensitivity has remained almost constant since the return
to the Earth. The average and standard deviation of the sensitivity after the
$\sigma$-clipping procedure is $16095\pm 490$ (DN/sec)/(W/m2/$\mu$m/sr) (red
solid and dashed lines, respectively, in Figure 10 left), which we applied to
all the data to obtain the sky brightness of the wide-band images. We treated
the standard deviation of the sensitivity as a systematic uncertainty (see
Section 4.3).

We applied the same sensitivity-calibration procedure to the v-band data
($V(x,y)/FLAT(x,y)$ images) to check the consistency of the result because the
wide-band data and v-band data share the same detector. Gaia does not have a
V-band filter, but it does have a blue band (BP) and a red band (RP), and the
V-band magnitude can be estimated from the BP and RP magnitudes [48]. In the
selection of stars for the v-band calibration, conditions (a) and (b) are the
same as for the wide-band calibration, and condition (c) selects stars
brighter than 8th mag in the converted V-band. Figure 10 (right) shows the
sensitivity profile of the v-band, and we confirmed that the sensitivity
remained constant in our dataset. The sensitivity and its 1$\sigma$
uncertainty we obtained for the v-band are $906\pm 37$
(DN/sec)/(W/m2/$\mu$m/sr).

Figure 11: Point-Spread Function. The PSF and its radial profile with 0.1
pixel resolution are shown on a linear scale (left) and a logarithmic scale
(right). Figure 12: Sky brightness. A histogram of the pixel signals of the
2022-10-17 data is shown on the vertical axis with a linear scale (left) and a
logarithmic scale (right). The red curves show the best-fit Gaussian functions
Table 2: Obtained sky brightness and backgrounds with their uncertainties.

Date | SKY(1) | ISL(1) | DGL(1) | ZL(1) | Corr. factor(2) | limmag(3)
---|---|---|---|---|---|---
2021-08-23 | 961.1 $\pm$ 23.2 $\pm$ 29.3 | 28.1 $\pm$ 2.2 | 34.2 $\pm$ 2.6 $\pm$ 27.6 | 889.3 $\pm$ 16.1 $\pm$ 40.3 | 1.142 | 12.79 $\pm$ 0.18
2021-09-20 | 1047.8 $\pm$ 6.0 $\pm$ 31.9 | 34.2 $\pm$ 2.7 | 33.2 $\pm$ 2.5 $\pm$ 25.9 | 958.4 $\pm$ 8.8 $\pm$ 41.1 | 1.135 | 12.80 $\pm$ 0.19
2021-11-29(4) | 1922.1 $\pm$ 12.1 $\pm$ 58.6 | 299.4 $\pm$ 15.1 | 46.9 $\pm$ 3.3 $\pm$ 30.3 | 1602.5 $\pm$ 17.4 $\pm$ 65.9 | 1.091 | 12.63 $\pm$ 0.16
2021-11-29(4) | 1937.1 $\pm$ 10.8 $\pm$ 59.0 | 299.1 $\pm$ 15.1 | 46.9 $\pm$ 3.3 $\pm$ 30.3 | 1619.5 $\pm$ 17.3 $\pm$ 66.3 | 1.091 | 12.63 $\pm$ 0.16
2021-12-06(4) | 1864.2 $\pm$ 9.8 $\pm$ 56.8 | 156.1 $\pm$ 8.4 | 32.5 $\pm$ 2.3 $\pm$ 22.1 | 1670.6 $\pm$ 11.3 $\pm$ 61.0 | 1.092 | 12.70 $\pm$ 0.17
2021-12-06(4) | 1819.9 $\pm$ 9.4 $\pm$ 55.4 | 155.8 $\pm$ 8.6 | 32.5 $\pm$ 2.3 $\pm$ 22.1 | 1642.8 $\pm$ 11.7 $\pm$ 59.7 | 1.091 | 12.70 $\pm$ 0.17
2021-12-28(4) | 1882.5 $\pm$ 10.1 $\pm$ 57.4 | 45.0 $\pm$ 3.1 | 24.4 $\pm$ 1.8 $\pm$ 18.6 | 1809.2 $\pm$ 7.9 $\pm$ 60.3 | 1.092 | 12.72 $\pm$ 0.17
2021-12-28(4) | 1885.6 $\pm$ 10.0 $\pm$ 57.4 | 45.7 $\pm$ 3.1 | 24.4 $\pm$ 1.8 $\pm$ 18.6 | 1770.7 $\pm$ 8.2 $\pm$ 60.4 | 1.092 | 12.71 $\pm$ 0.17
2022-04-18 | 1599.7 $\pm$ 11.0 $\pm$ 48.7 | 186.3 $\pm$ 8.3 | 68.7 $\pm$ 5.0 $\pm$ 49.1 | 1358.8 $\pm$ 12.3 $\pm$ 69.2 | 1.172 | 12.69 $\pm$ 0.17
2022-05-16 | 1052.9 $\pm$ 18.9 $\pm$ 32.1 | 57.0 $\pm$ 3.8 | 16.7 $\pm$ 1.3 $\pm$ 13.0 | 988.3 $\pm$ 9.3 $\pm$ 34.6 | 1.202 | 12.76 $\pm$ 0.18
2022-06-20 | 857.3 $\pm$ 32.7 $\pm$ 26.1 | 31.8 $\pm$ 2.4 | 15.1 $\pm$ 1.2 $\pm$ 12.2 | 806.1 $\pm$ 13.9 $\pm$ 28.8 | 1.198 | 12.74 $\pm$ 0.18
2022-07-04 | 928.0 $\pm$ 9.4 $\pm$ 28.3 | 27.9 $\pm$ 2.1 | 25.7 $\pm$ 2.0 $\pm$ 20.9 | 850.2 $\pm$ 8.0 $\pm$ 35.2 | 1.190 | 12.77 $\pm$ 0.18
2022-08-01 | 994.2 $\pm$ 15.2 $\pm$ 30.3 | 31.7 $\pm$ 2.4 | 17.2 $\pm$ 1.3 $\pm$ 13.7 | 927.5 $\pm$ 8.2 $\pm$ 33.2 | 1.172 | 12.79 $\pm$ 0.18
2022-08-29 | 1196.4 $\pm$ 12.8 $\pm$ 36.4 | 38.7 $\pm$ 2.7 | 132.9 $\pm$ 9.9 $\pm$ 99.2 | 1002.4 $\pm$ 14.2 $\pm$ 105.7 | 1.147 | 12.79 $\pm$ 0.18
2022-10-17 | 1984.2 $\pm$ 9.4 $\pm$ 60.4 | 251.7 $\pm$ 12.4 | 45.2 $\pm$ 3.2 $\pm$ 29.6 | 1698.3 $\pm$ 14.6 $\pm$ 67.3 | 1.102 | 12.62 $\pm$ 0.16
2022-11-14 | 1854.4 $\pm$ 9.3 $\pm$ 56.5 | 44.5 $\pm$ 2.9 | 20.0 $\pm$ 1.5 $\pm$ 15.2 | 1792.9 $\pm$ 7.6 $\pm$ 58.5 | 1.114 | 12.69 $\pm$ 0.17
2022-12-12 | 1997.9 $\pm$ 7.8 $\pm$ 60.9 | 37.0 $\pm$ 2.6 | 23.6 $\pm$ 1.8 $\pm$ 18.8 | 1923.5 $\pm$ 6.8 $\pm$ 63.7 | 1.121 | 12.70 $\pm$ 0.17

  * 1

(1) Brightness in $\lambda I_{\lambda}$ and its statistical and systematic
uncertainty in nW/m2/sr.

  * 2

(2) Correction factor used to obtain the ZL brightness toward the antisolar
direction in the ecliptic plane (see Section 4.2).

  * 3

(3) Limiting magnitude in the G-band (see Section 2.10).

  * 4

(4) Two data sets were acquired on the same day (see Section 2.3).

###  2.8 Point-Spread Function

We obtained a template for the point-spread function (PSF) of the ONC-T wide-
band image by adding the images of several bright stars. First, as suitable
images for making the PSF template, we selected 56 objects with G-band
magnitudes between 6th mag and 8th mag that have clean stellar images, with
few bad pixels or without other objects around them. Then, we aligned these
images with 0.1 pixel resolution and added them together to obtain the PSF
template. Figure 11 shows the resulting PSF template and its radial profile
with 0.1 pixel resolution. The full-width half maximum (FWHM) of the PSF is
2.00 pixels, which is consistent with previous measurements [56].

###  2.9 Sky Brightness

We obtained the sky brightness as the signal value of dark pixels with no
stars nor hot pixels by cosmic-ray hits in the $SKY(x,y)$ image, using the
histogram method employed to determine the dark current described in Section
2.4. We created a histogram of the $SKY(x,y)$ image, including stars and hot
pixels, and obtained the best-fit Gaussian curve. Figure 12 shows the
histogram of the $SKY(x,y)$ image and its best-fit Gaussian curve for the
2022-10-17 data. Since the $SKY(x,y)$ image is dominated by dark pixels with
no stars nor hot pixels, the peak of the pixel histogram of the $SKY(x,y)$
image represents the sky brightness. We separate the higher signal tail of
bright stars and the dark sky signals in the pixel histogram by the Gaussian
fitting as shown in Figure 12, and we treated the 1$\sigma$ error in the peak
position as the statistical uncertainty in the sky brightness. We converted
the resulting sky brightness from DN units to $\lambda I_{\lambda}$ in
nW/m2/sr units by applying the calibration factor obtained in Section 2.7.
Using this procedure, the systematic uncertainty in the calibration factor is
transferred to a systematic uncertainty in the sky brightness. Table 2
summarizes the obtained sky brightness and its statistical and systematic
uncertainties.

At this point, the detection limit has not been determined (this is obtained
in the next subsection using the sky brightness and its standard deviation).
As it is not known how faint stars should be masked based on the star catalog,
we determined the sky brightness by creating a histogram of the entire
$SKY(x,y)$ image without masking the stars. This method worked well because
the number of pixels observing dark sky is much larger than the number
observing stars and other astronomical objects. Note that the stars detected
in the images are masked when we determine the ZL as described in Section 3.4.

###  2.10 Limiting Magnitude

It is important to know the limiting magnitude in our observed images because
we need to mask the detected stars to derive the diffuse brightness of the
sky. As shown in Figure 12, the histogram of the $SKY(x,y)$ image has a side
lobe caused by the detected stars in the images, which exceeds the best-fit
Gaussian curve. The width ($\sigma$) of the Gaussian corresponds to the
standard deviation of the fluctuation in the sky brightness, and we define the
limiting magnitude as the brightness of stars for which three pixels in the
center of the PSF (Figure 11) exceed $+2\sigma$ of the sky deviation. We set
the uncertainty in determining the limiting magnitude to be 1 DN, which is
equivalent to approximately 0.2 mag uncertainty in the limiting magnitude. The
limiting magnitude of each image and its uncertainty are summarized in Table
2.

Stars brighter than the limiting magnitude were extracted from the Gaia DR3
catalog (see Section 2.7), convolved with the PSF (see Section 2.8), and
distributed in the image to create a Gaia bright-star image, as shown in
Figure 13 (b). The distribution of stars in this Gaia bright-star image
reproduces well the distribution of the stars detected in the image observed
by ONC-T (Figure 13 (a)), indicating the validity of the detection limits
determined by the method described above. We used the Gaia bright-star image
as a stellar mask to conceal stars when obtaining the ZL (see Section 3.4).

Figure 13: SKY(x,y) image obtained by ONC-T and ISL image. (a): A zoom-in
image taken by ONC-T. (b): Gaia bright-star image of the same area as panel
(a). We obtained this image using stars brighter than the limiting magnitude
of the Gaia DR3 catalog, and we used this as a mask to conceal the stars
detected in the SKY(x,y) image. (c): Gaia ISL image of the same area as panel
(a). We constructed this image using stars fainter than the limiting magnitude
of the Gaia DR3 catalog. The intensity scale of this image (c) is several
times larger because the stars are too faint to be seen at the same scale.
(d): Masked image of SKY(x,y) shown in (a) obtained using the mask image (b).

##  3 Background Subtraction

###  3.1 Integrated Starlight

Stars fainter than the limiting magnitude are not detected as point sources in
the observed images, but the sum of the light from those undetected stars,
called integrated starlight (ISL), contributes to the sky brightness.
Therefore, the ISL must be estimated and subtracted from the sky brightness to
obtain the ZL.

The ISL image $ISL(x,y)$ for each field is produced by stars fainter than the
limiting magnitude extracted from the Gaia DR3 catalog, as shown in Figure 13
(c). The average brightness of each ISL image and its $1\sigma$ statistical
uncertainty are listed in Table 2. Since the ISL at optical wavelengths
saturates when the contributions from stars down to 20th mag are added [10],
the depth of the Gaia DR3 catalog is sufficient. The photometric uncertainty
in the catalog is negligible compared with the uncertainty of the limiting
magnitude of ONC-T.

###  3.2 Diffuse Galactic Light

Diffuse Galactic light (DGL) consists of starlight scattered by interstellar
dust in our Galaxy [63], and it also must be subtracted from the sky
brightness to obtain the ZL. A method commonly used to estimate the DGL is to
use its correlation with the thermal emission from interstellar dust in the
far infrared. The intensity map at $\lambda=$100 $\mu$m, which is a
reprocessed composite of the COBE and IRAS maps (SFD map [64]), is commonly
used as a template for the interstellar dust distribution. Thus,

$\lambda I_{DGL}=\nu\beta_{\lambda}\cdot d(Glat)\cdot I_{SFD},$ (6)

where $\lambda I_{DGL}$ is the DGL brightness in nW/m2/sr, $I_{SFD}$ is the
far-infrared intensity at 100 $\mu$m from the SFD map in MJy/sr, $d(Glat)$ is
a geometric function of the Galactic latitude ($Glat$), and
$\nu\beta_{\lambda}$ is the DGL correlation factor in (nW/m2/sr)/(MJy/sr). As
discussed in [65], this geometric function is given by

$d(Glat)=d_{0}(1-1.1g\sqrt{\sin|Glat|}),$ (7)

where $d_{0}$ is a normalizing parameter, and $g$ is the asymmetry factor of
the scattering phase function [66].

This DGL correlation factor at optical and near-infrared wavelengths has been
derived in many previous studies [67, 68, 69, 70, 71, 72, 73, 74, 75], but
there are two different results. Recently, the values
$\nu\beta_{\lambda}=3.54\pm 0.91$ (nW/m2/sr)/(MJy/sr), $d_{0}=1.76$, and
$g=0.61$ have been reported from results obtained by the Long-Range
Reconnaissance Imager (LORRI) on New Horizons [75]. These results were
obtained at 10-50 au from the Sun, where the ZL is negligible. The bandpass of
LORRI is also similar to that of the wide-band filter of ONC-T (Figure 1).
However, this DGL estimate is about 5-10 times smaller than many previous
results. For example, $\nu\beta_{\lambda}=21.0\pm 0.9$ [(nW/m2/sr)/(MJy/sr)]
and $d(Glat)=1$ (the geometric function was not considered) at 0.65 $\mu$m was
found based on observations from the Hubble Space Telescope (HST) [73]. In the
present work, we treat the DGL estimate based on the New Horizons result as a
low-level DGL estimate, that based on the HST result as a high-level DGL
estimate, and the average of these two DGL estimates as a middle-level DGL
estimate. We treat the difference between the low-level and the high-level DGL
estimates as a systematic uncertainty.

Figure 14: Diffuse Galactic Light. Left: DGL image for the field of 2021-08-23
based on the SFD map at $\lambda=$100 $\mu$m [64]. Center: DGL image for the
same field based on the AKARI Wide-S map at $\lambda=$90 $\mu$m [76, 77] after
masking the point sources. Right: Comparison of the intensities from the SFD
map and the AKARI Wide-S map of our observation fields. The red line shows the
best fit to this correlation.

The spatial resolution of the SFD map (6.1 arcmin) is insufficient relative to
our data from ONC-T (22 arcsec). Therefore, we used a far-infrared, all-sky
diffuse map based on the AKARI all-sky survey [76, 77] in this study. Because
these AKARI all-sky diffuse maps cover wider wavelength ranges with finer
spatial resolution and better signal-to-noise ratio than the SFD map, they can
serve as a new template for the DGL estimate, replacing the SFD map. In this
study, we used the AKARI Wide-S ($\lambda=$90 $\mu$m) map, for which the
spatial resolution is $\sim 1.3$ arcmin. One difference between these maps is
that point sources have not been removed from the AKARI Wide-S map, whereas
they have been removed from the SFD map. Therefore, we masked all the point
sources included in the AKARI Far-Infrared Bright Source Catalogue Version 2
[78]. Figure 14 compares the SFD map (left) and AKARI Wide-S map (center) in
the field of 2022-08-23, which shows that the AKARI Wide-S map has better
spatial resolution than the SFD map

The publicly available AKARI Wide-S map has a ZL remainder that must be
subtracted because only the smooth cloud component of the ZL has been
subtracted from the raw data, and other ZL components, such as asteroidal dust
bands, have not been subtracted [76]. The contribution from the unsubtracted
ZL components is recognizable in the Wide-S map in the low-ecliptic-latitude
region that we study in this work. There is a good linear correlation between
the SFD map and the AKARI Wide-S map [77], but the AKARI Wide-S map at low
ecliptic latitudes shows deviations from the linear correlation owing to the
residual ZL that remains to be subtracted. Therefore, we used the data that
other ZL components are additionally subtracted from the public AKARI Wide-S
map based on a ZL asteroidal-dust-band model [79]. We confirmed the good
correlation between the SFD map and the additionally ZL-subtracted AKARI
Wide-S map, as shown in Figure 14 (right). This correlation is fitted by the
equation

$I_{\textrm{{\small SFD}}}=a\times I_{\textrm{{\small Wide-S}}}+c$ (8)

where $I_{\textrm{{\small Wide-S}}}$ is the far-infrared intensity from the
AKARI Wide-S map in MJy/sr, and $a$ and $c$ are the fitting parameters. We
obtained $a=1.54\pm 0.05$ and $c=-1.20\pm 0.14$ MJy/sr in our observation
fields, which are consistent with the result based on the all-sky data [76,
77]. We converted the AKARI Wide-S maps in our observation fields into DGL
images $DGL(x,y)$ using equations (6) and (8).

Another issue in the AKARI Wide-S map is the sky coverage. The AKARI all-sky
map has $>99\%$ coverage of the whole sky, but our observation fields contain
regions with missing data. We therefore used SFD data for the missing regions
in the AKARI Wide-S map. The middle-level DGL brightness based on the AKARI
Wide-S map and its statistical and systematic uncertainties are listed in
Table 2.

###  3.3 Extragalactic Background Light

Extragalactic background light (EBL) arises from emissions integrated from the
first era of star production to the present day. Recent observations have
shown that the EBL measured at optical and near-infrared wavelengths has an
excess over the cumulative light from galaxies [80, 81, 82, 83, 84, 85, 86,
87, 75, 88, 89], which means that there are unknown light sources in the
universe. The sources for this excess are still under discussion, but some
candidates that have been proposed include intra-halo light [90, 91],
primordial black holes formed by the collapse of the first halos [92], the
decay of hypothetical particles [93], nearby black holes observed as faint
compact objects [94, 95], and a warm-hot intergalactic medium [96]. We adopted
$\lambda I_{EBL}=21.98\pm 1.83$ nW/m2/sr at $\lambda$ = 0.44-0.87 $\mu$m, as
observed by LORRI/New Horizons [75]. We created an EBL image $EBL(x,y)$ with
all pixels having this value.

Figure 15: A histogram of the pixel signals from the masked ZL image of the
2022-10-17 data on a linear scale (left) and a logarithmic scale (right). The
red curves show the best-fit Gaussian functions, and the red dashed line shows
the $+2\sigma$ distance from the peak position. We performed the fitting using
data within this range.

###  3.4 Zodiacal Light

The ZL is obtained by subtracting the background emissions from the observed
sky brightness:

$ZL(x,y)=SKY(x,y)-ISL(x,y)-DGL(x,y)-EBL(x,y).$ (9)

We first subtracted the ISL image (Section 3.1), DGL image (Section 3.2), and
EBL image (Section 3.3) from the SKY image (Section 2.6) to obtain the ZL
image, $ZL(x,y)$, and we masked the stars detected in the ZL image (Section
2.10). As shown in Figure 13 (d), this masking procedure works well for almost
all stars, although the peripheries of some of the brightest stars are not
masked perfectly and are smeared out. We then created a histogram of the area
where $FLAT(x,y)$ $>0.5$ and stray light $<20$ DN (see Section 2.7) for each
masked ZL image, and we regard its peak position and its 1$\sigma$ error
estimate as the brightness and statistical uncertainty of the ZL (Figure 15).
The systematic uncertainty in the ZL comes from the systematic uncertainties
in the calibration and the DGL. Comparing the histograms of the masked ZL
image (Figure 15) with the SKY image (Figure 12) shows that the excess of the
side lobe over the Gaussian has been reduced thanks to the stellar masking.
There is still a small excess owing to the smeared peripheries of the
brightest stars and to some hot pixels caused by cosmic-ray hits, but this
small excess has little effect on the peak position of the Gaussian because we
performed the Gaussian fitting on data within a $+2\sigma$ range from the peak
(the red dashed line in Figure 15). The resulting estimate of the ZL and its
uncertainties are listed in Table 2 and shown in Figure 16 (black).

##  4 Discussion

###  4.1 Absolute ZL brightness

We have compared the ZL brightness we observed with that predicted by the
Kelsall model, which is based on observations of the all-sky ZL brightness
obtained from COBE observations at infrared wavelengths [23]. We calculated
the ZL model brightness using the ZodiPy code [97], which implements the
Kelsall model. The shortest wavelength for which the ZL brightness can be
calculated with this model is 1.25 $\mu$m, which is outside the range of the
ONC-T wide-band data used in this study. However, the ZL at both 1.25 $\mu$m
and optical wavelengths (i.e., the ONC-T wide-band filter) results from
scattered sunlight, and its spectral shape is the same irrespective of
ecliptic latitude [5]. We therefore determined the ZL brightness at optical
wavelengths by extrapolating from the model brightness at 1.25 $\mu$m using
the solar spectrum [98]. This extrapolation was performed using the ratio of
the solar spectrum at 0.612 $\mu$m and 1.25 $\mu$m. Since the IDP reflectance
varies by about 10% between 1.25 $\mu$m and optical wavelengths [5, 84, 99],
we have assumed a 10% uncertainty in the ZL model brightness at optical
wavelengths associated with this extrapolation. Figure 16 compares the
observed ZL brightness with the model brightness, which shows that they are
consistent with each other within the ranges of uncertainties.

Figure 16: The ZL brightness of each field (black). The red points show the
model brightness [23] extrapolated to optical wavelength using the solar
spectrum [98]. Figure 17: Field-variance correction. Left: ZL model brightness
of the fields observed by Hayabusa2# (black) and toward the antisolar
direction in the ecliptic plane (red) based on the Kelsall model [23] at 1.25
$\mu$m. A radial power-law profile with $\alpha=1.34$ is also shown. Right:
Observed ZL brightness after subtracting the background components (black) and
the corrected ZL brightness toward the antisolar direction in the ecliptic
plane (red).

Gegenschein appears in the antisolar direction, and the Kelsall model does not
include it. However, our observed fields are shifted from the antisolar
direction by $\sim$10 degrees (see Table 1), and the Gegenschein is negligible
there [100].

There are small excesses of the observed ZL brightness over the model
brightness at around 0.9 au, as shown in Figure 16. This structure may be
real, but we cannot confirm this at this point due to the paucity of data
points. Verification will require the accumulation of data from future
observations.

###  4.2 Field-Variance Correction

Since the ZL brightness measurements were obtained at different ecliptic
latitudes and solar elongations, a correction for the field variance is
necessary to compare them under the same conditions in order to obtain the
radial profile of the ZL. We performed this field-variance correction based on
the Kelsall model. We calculated the seasonal average of the ZL model
brightness toward the antisolar direction in the ecliptic plane at various
heliocentric distances, as shown in Figure 17 (left, red). The radial power-
law index of this calculated ZL model brightness is $\alpha=1.34$ because this
value is used in the Kelsall model. We compared this ZL brightness toward the
antisolar direction with the ZL model brightness toward the fields observed by
ONC-T (Figure 17, left, black). We multiplied the observed ZL brightness by
the ratio of these two ZL model brightnesses at each position as correction
factors in order to obtain the ZL brightness toward the antisolar direction in
the ecliptic plane (Figure 17 right). These correction factors are listed in
Table 2.

Since this correction relies on the Kelsall model with $\alpha=1.34$, it is
not self-consistent if the obtained value of $\alpha$ deviates significantly
from 1.34. As we discuss in the next subsection, however, the value of
$\alpha$ we obtained from our observations is close to 1.34, so consequently
this correction works. In addition, the field variance was corrected for local
brightness difference at the same elongations but different ecliptic
coordinates, so the correction is not sensitive to $\alpha$ which represents
the global distribution of IPD.

Table 3: Obtained radial power-law index $\alpha$ for each systematic uncertainty case. Calibration | DGL | Value of $\alpha$
---|---|---
low | low | $1.19\pm 0.02$
nominal | low | $1.22\pm 0.02$
high | low | $1.21\pm 0.02$
low | middle | $1.29\pm 0.02$
nominal | middle | $1.28\pm 0.02$
high | middle | $1.31\pm 0.02$
low | high | $1.38\pm 0.02$
nominal | high | $1.37\pm 0.02$
high | high | $1.42\pm 0.02$
Figure 18: The radial profile of the ZL observed by Hayabusa2# on 2021-2022 at
0.76-1.06 au on a linear scale (left) and a logarithmic scale (right). The
error bars include both statistical and systematic uncertainties. The solid
curve shows the best-fit power-law function ($\alpha=1.30$), and the dashed
curves show the power-law functions for the low-DGL ($\alpha=1.19$) and high-
DGL ($\alpha=1.42$) cases.

###  4.3 Dependence on Heliocentric Distance

Based on the corrected ZL brightness, we calculated the radial power-law index
$\alpha$ using the following method. The ZL brightness has two types of
uncertainties: statistical uncertainties and systematic uncertainties.
Statistical uncertainties appear randomly at each data point, while systematic
uncertainties appear with a certain tendency at each data point. Our data
contain two types of systematic uncertainties, one due to calibration (see
Section 2.7) and the other due to DGL (see Section 3.2). We therefore
calculated the radial power-law index $\alpha$ for a total of $3\times 3$
cases (three cases for calibration uncertainty and three cases for DGL
uncertainty). Table 3 shows the values of $\alpha$ we obtained for each of
these systematic-uncertainty cases. As this table shows, the calibration
uncertainty does not have a significant impact on the value of $\alpha$, since
the data points only go up and down overall. On the other hand, the DGL
uncertainty does have a significant impact on the value of $\alpha$. From all
of these values, we obtain $\alpha=1.30\pm 0.08$ as the final result. Figure
18 shows the radial profile of the ZL and the best-fit power-law function.
Again, the excess structure at $\sim$0.9 au can be seen in Figure 18.

Table 4: Comparison of the radial power-law index $\alpha$.

Value of $\alpha$ | Coverage of $\alpha$ | Method | Observation wavelength | Observation site | Instrument | Reference
---|---|---|---|---|---|---
$1-1.5$ |  $1-3.3$ au | ZL observation | B, R bands | Interplanetary space | Pioneer 10/11 | [33]
$1.3\pm 0.05$ |  $0.3-1$ au | ZL observation | U, B, V bands | Interplanetary space | Helios 1/2 | [34]
$1.34\pm 0.022$ |  $>1$ au | ZL observation | 1.25-240 $\mu$m | Geocentric orbit | COBE | [23]
$1.22$ |  $>1$ au | ZL observation | 1.25-240 $\mu$m | Geocentric orbit | COBE | [24]
$1.45\pm 0.05$ |  $0.06-0.6$ au | F-corona | 0.5-0.9 $\mu$m | Lunar orbit | Clementine | [38]
$1.59\pm 0.02$ |  $>1$ au | ZL observation | 9 $\mu$m, 18 $\mu$m | Geocentric orbit | AKARI | [37]
$1.34^{(1)}$ |  $1-3.3$ au | ZL observation | B, R bands | Interplanetary space | Pioneer 10/11 | [99]
$1.31-1.35$ |  $0.07-0.45$ au | F-corona | 0.63-0.73 $\mu$m | Heliocentric orbit | STEREO-A | [39]
$1.31$ |  $0.1-0.4$ au | F-corona | 0.49-0.74 $\mu$m | Interplanetary space | PSP | [41]
$2$ |  $0.17-0.7$ au | Dust counting | — | Interplanetary space | PSP | [29]
$1.30\pm 0.08$ |  $0.76-1.06$ au | ZL observation | 0.39-0.84 $\mu$m | Interplanetary space | Hayabusa2# | This work

  * 1

(1) the Kelsall model is assumed.

Figure 19: Comparison of the radial power-law index $\alpha$ as a function of
heliocentric distance. The data plotted here are shown in Table 4. The red
data point shows the result from this study.

Table 4 and Figure 19 compare the value of $\alpha$ we obtained with previous
results, and they show that our result is consistent with them. Since it is
difficult to determine the radial profile of the IDP density from ZL
observations obtained in a geocentric orbit or from F-corona observations,
direct observations of the radial profile of the ZL from interplanetary space
outside Earth’s orbit are more reliable for this purpose. Our observations are
the first successful observations of ZL from interplanetary space in the 40
years since Helios 1/2 and Pioneer 10/11. In addition, the ZL intensity varies
with both the heliocentric distance and the solar elongation, both of which
varied in the previous observations by Helios 1/2 and Pioneer 10/11. On the
other hand, we confined our observations to the antisolar direction (solar
elongation $\sim$180 deg), so we can impose changes in ZL brightness on
changes in heliocentric distance.

Values of $\alpha$ greater than 1 are obtained from all the ZL observations,
even though $\alpha=1$ is expected if the orbital evolution of the IDP is
dominated by the PR effect [2]. This difference is caused by dust production
due to the collision of dust particles [101, 43], dust supplied by comets
around 1 au [102], the finiteness of the dust cloud [103], or the heliocentric
dependence of the local albedo of the IDP [104, 32]. In fact, a heliocentric
dependence of the local albedo of the form $r^{-0.3\pm 0.1}$ has been reported
[32], which partially explains $\alpha$ being greater than 1.

The IDPs falling into the Sun due to the PR effect decrease in size owing to
evaporation, and such small IDPs are blown away by radiation pressure as
$\beta$-meteoroids. The radial profile of $\beta$-meteoroids is expected to
follow a power law with $\alpha=2$, and recent in-situ direct counting of the
flux of IDPs experienced by the PSP does show $\alpha=2$ in the range
$0.17-0.7$ au [29]. On the other hand, the reddening [5] and polarization [20,
21] of the ZL spectrum indicate that the majority of IDPs seen as ZL are large
($>1$ $\mu$m); smaller IDPs do not contribute much to the ZL even though they
do exist [28]. While the value $\alpha=2$ obtained by the IDP impact-counting
method is sensitive to small IDPs, the values of $\alpha$ between 1 and 2
obtained from ZL observations are sensitive to larger IDPs. The small dust
particles become hotter than the larger ones [105], and a hot component in the
thermal emission from IDPs has been found using mid-infrared spectroscopy at
$\lambda=$ 3-6 $\mu$m [106, 107, 108, 17]. We would therefore expect to obtain
the value $\alpha\sim 2$ from ZL observations at $\lambda=$ 3-6 $\mu$m carried
out outside Earth’s orbit because small dust particles with high temperatures
are mainly observed in this wavelength range.

###  4.4 Future Observations

Since ZL observations at 0.7-1 au by Hayabusa2# will continue until the Earth
swing-by at the end of 2027, the accuracy of the results we have reported here
will be improved through the accumulation of additional observational data. In
particular, the excess structure at $\sim$0.9 au needs to be verified by
accumulating data during this phase. After the second Earth swing-by in 2028,
the Hayabusa2 spacecraft will fly to an orbit in the 1-1.5 au range [52], so
we will be able to obtain the radial profile in the outer regions of the Solar
System.

Before this orbital change of Hayabusa2#, we will have a chance to observe the
radial profile of the ZL in the 1-1.5 au range from the Martian Moons
eXploration (MMX) spacecraft, which is a Japanese sample-return mission from
the Martian satellite Phobos [109]. The MMX is scheduled for launch in 2024
and arrival at Mars in 2025, and we are proposing to conduct simultaneous
multi-wavelength (350-1000 nm) ZL observations during this cruising phase
using the Optical RadiOmeter composed of Chromatic Imagers (OROCHI) onboard
MMX [110].

In addition, we are developing an EXo-Zodiacal Infrared Telescope (EXZIT),
with the aim of installing it on a spacecraft to Jupiter or farther [111,
112]. If this instrument can be realized, we will be able to observe the
radial profile of the ZL at 1-5 au as well as the EBL without the ZL
foreground above 3 au. We are also considering adding mid-infrared
capabilities to EXZIT, which would allow us to examine our prediction of the
$\alpha\sim 2$ index for the ZL radial profile owing to small particles.

In-situ direct dust counting is also important for comprehending the IDP
distribution in the Solar System because it provides independent estimates of
the radial variation of the IDP density. More quantitative comparisons between
the ZL observations and in-situ direct dust counting are envisioned for future
projects, which will provide us with the differences in the distributions
according to dust size and parent bodies.

##  5 Summary

We observed the ZL brightness at optical wavelengths at 0.76-1.06 au with
ONC-T on the Hayabusa2# mission. We detected a small excess of the observed ZL
brightness over the model brightness at around 0.9 au, but we cannot determine
whether or not this structure is real at this stage due to the paucity of data
points. The radial power-law index we obtained is $\alpha=1.30\pm 0.08$, and
the uncertainty in this estimate is dominated by the uncertainty due to the
DGL estimate. This result is consistent with previous results based on other
ZL observations.

## Acknowledgements

The Hayabusa2 spacecraft was developed and built under the leadership of the
Japan Aerospace Exploration Agency (JAXA), with contributions from the German
Aerospace Center (DLR) and the Centre National d’études Spatiales (CNES), and
in collaboration with NASA, Nagoya Univ., The Univ. of Tokyo, National
Astronomical Observatory of Japan (NAOJ), Univ. of Aizu, Kobe Univ., and other
universities, institutes, and companies in Japan. This work has made use of
the data obtained with AKARI, a JAXA project with the participation of the
European Space Agency (ESA). This work also has made use of data from the 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 research has made use
of the VizieR catalog access tool, CDS, Strasbourg, France (DOI :
10.26093/cds/vizier). The original description of the VizieR service was
published in [113]. The authors thank Takafumi Ootsubo (NAOJ) for providing
the ZL-subtracted AKARI Wide-S data, and Teresa Symons (University of
California, Irvine) and Michael Zemcov (Rochester Institute of Technology) for
discussion on DGL estimate based on New Horizons.

## Funding

K. Tsumura was supported by JSPS KAKENHI Grant Number 20H04744 and Tokyo City
University Prioritized Studies.

## Abbreviations

CNES: Centre National d’études Spatiales; COBE: Cosmic Background Explorer;
DARTS: Data Archives and Transmission System; DGL: Diffuse Galactic light; DN:
digital number; DPAC: Data Processing and Analysis Consortium; DR: Data
release; EBL: Extragalactic background Light; ESA: European Space Agency;
EXZIT: Exo-Zodiacal Infrared Telescope; FWHM: Full-width half maximum; HST:
Hubble Space Telescope; IDP: Interplanetary dust particles; ISL: Integrated
star light; JAXA: Japan Aerospace Exploration Agency; LORRI: Long-Range
Reconnaissance Imager; MMX: Martian Moons eXploration; NAOJ: National
Astronomical Observatory of Japan; ONC: Optical Navigation Camera; OROCHI:
Optical RadiOmeter composed of CHromatic Imagers; PR effect: Poynting-
Robertson effect; PSF: Point spread function; PSP: Parker Solar Probe; STEREO:
Solar TErrestrial RElations Observatory; WISPER: Widefield Imager for Solar
Probe inner telescope; ZL: Zodiacal light.

## Availability of data and materials

The raw datasets of this study will be made available in the Hayabusa2 Science
Data Archives on the Data Archives and Transmission System (DARTS) repository,
at https://www.darts.isas.jaxa.jp/planet/project/hayabusa2/. The analyzed
datasets of this study are available from the corresponding author upon
reasonable request.

## Competing interests

No competing interest is declared.

## Authors’ contributions

K.Ts. analyzed the data and wrote the manuscript. S.M., K.Sa., T.I., and K.Ta.
contributed to the data analysis. M.Y., T.M., T.K., M.H., Y.Y., E.T., M.M.,
N.S., R.H., S.K., H.Su., K.Y., Y.C., K.O., K.Sh., H.Sa., and S.S. contributed
to ONC data acquisitions and reduction. M.Y. contributed to preparing the
operation sequence for the ZL observations. All authors discussed the results
and approved the final manuscript.

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