# The Geometric Albedo of (4179) Toutatis Estimated from
KMTNet DEEP-South Observations

Yoonsoo P. Bach  Department of Physics and Astronomy, Seoul National
University, 1 Gwanak, Seoul 08826, Korea;  Masateru Ishiguro  Department of
Physics and Astronomy, Seoul National University, 1 Gwanak, Seoul 08826,
Korea;  Sunho Jin  Department of Physics and Astronomy, Seoul National
University, 1 Gwanak, Seoul 08826, Korea;  Hongu Yang  Korea Astronomy and
Space Science Institute, 776, Daedeok-daero, Yuseong-gu, Daejeon 34055, Korea
Hong-Kyu Moon  Korea Astronomy and Space Science Institute, 776, Daedeok-
daero, Yuseong-gu, Daejeon 34055, Korea  Young-Jun Choi  Korea Astronomy and
Space Science Institute, 776, Daedeok-daero, Yuseong-gu, Daejeon 34055, Korea
Youngmin JeongAhn  Korea Astronomy and Space Science Institute, 776, Daedeok-
daero, Yuseong-gu, Daejeon 34055, Korea  Myung-Jin Kim  Korea Astronomy and
Space Science Institute, 776, Daedeok-daero, Yuseong-gu, Daejeon 34055, Korea
SungWon Kwak  Korea Astronomy and Space Science Institute, 776, Daedeok-daero,
Yuseong-gu, Daejeon 34055, Korea<EMAIL_ADDRESS>
(Received January 25, 2019; accepted May 21, 2019)

##  1 Introduction

The geometric albedo is one of the most fundamental observed quantities for
characterizing the physical properties and composition on asteroids. It
provides information on the surface composition, degree of space weathering,
and physical conditions such as the surface particle size if polarimetric data
at large phase angles (Sun–asteroid–observer angle) are available (Shkuratov &
Opanasenko, 1992; Dollfus, 1998). Because the absolute photometric magnitude
(at the opposite direction from the Sun viewed at unit heliocentric and
observer distances) is related to the product of the geometric albedo and
cross section of the asteroid, the accuracy of the size determination is a
critical factor for deriving the geometric albedo in most cases. The sizes of
asteroids have been investigated thoroughly via various techniques, namely,
direct imaging of asteroids with cameras onboard spacecraft, delay-Doppler
imaging (Ostro, 1993), thermal–infrared observations (Tedesco et al., 2002;
Mainzer et al., 2011; Usui et al., 2011), and occultation (Shevchenko &
Tedesco, 2006); nevertheless, absolute magnitudes are rarely studied because
of the scarcity of observational opportunities at opposition. In addition,
magnitude variations due to an asteroid’s rotation have not been considered in
many cases for the derivation, making the albedo values less certain. For
these reasons, the number of reliable albedo estimates for asteroids is
limited.

Here, we examine the geometric albedo of an Apollo-type near-Earth asteroid
(NEA), (4179) Toutatis. The asteroid has been thoroughly studied by various
observational techniques (i.e., photometry, spectroscopy, polarimetry, radar
imaging, and in situ observation). It is classified into S-type (Lazzarin et
al., 1994; Howell et al., 1994) or, more strictly, $\mathrm{S_{k}}$-type (Bus
& Binzel, 2002) and is possibly composed of undifferentiated L-chondrites
(Reddy et al., 2012). Its physical dimensions are
$4.6\mathrm{\,\mathrm{km}}\times 2.4\mathrm{\,\mathrm{km}}\times
1.9\mathrm{\,\mathrm{km}}$ according to radar observations (Hudson & Ostro,
1995). Toutatis is in an extremely slow, non-principal-axis rotational state
(Ostro et al., 1995; Takahashi et al., 2013) having rotation and precession
periods of $\sim 5.38\mathrm{\,\mathrm{days}}$ and $\sim
7.40\mathrm{\,\mathrm{days}}$, respectively (Zhao et al., 2015). An accurate
shape model was established by radar delay-Doppler imaging (Hudson et al.,
2003).

More recently, in December 2012, the Chinese spacecraft Chang’e-2 captured
images of this asteroid with a maximum resolution better than
3$\mathrm{\,\mathrm{m}}$ during the flyby (Huang et al., 2013). The study
revealed, e.g., a previously unknown $\sim 805\mathrm{\,\mathrm{m}}$ basin at
the end of the body, lineaments over the surface especially near the rim of
it, and a sharp connection silhouette in the neck, as well as providing
valuable information on the asteroid’s size ($4.75\times
1.95\mathrm{\,\mathrm{km}}$ with nominal uncertainty of 10%). Moreover, more
than 50 craters larger than 36$\mathrm{\,\mathrm{m}}$ and 30 boulders larger
than 10$\mathrm{\,\mathrm{m}}$ were identified and the chronology of craters
and size distribution of boulders were investigated. From the cumulative
boulder size frequency distribution analyses compared with (25143) Itokawa, it
has been suggested that Toutatis may have a different preservation state or a
diverse formation history (Jiang et al., 2015). The shape is considered to
have been produced by a low-speed impact between two components (Hu et al.,
2018). The hemispherical albedo was found (Zhao et al., 2016) to be 0.2083,
0.1269, and 0.1346 in the R, G, and B bands of the Chang’e-2 CMOS censor,
respectively. Moreover, the close-orbit dynamics was used to explore Toutatis’
complex gravitational field. The asteroid is also a notable for its extremely
chaotic orbit in a 3:1 mean motion resonance with Jupiter and a weak 1:4
resonance with Earth (Whipple & Shelus, 1993). In addition, the asteroid has a
unique potential field in the close orbit (Scheeres et al., 1998). Thus,
Toutatis has been central to the developments in NEA research since the 1990s.

Although Toutatis has been studied intensively, its geometric albedo is
uncertain by a factor of $>3$, ranging from 0.13 (Lupishko et al., 1995) to
0.41 (Masiero et al., 2017). In this paper, we aim to establish a better
estimate of the albedo value. Taking advantage of the continuous observation
capability from KMTNet observatories in three different continents (Kim et
al., 2016, see Section 2), we derived the geometric albedo of the slowly
rotating asteroid. Together with the polarimetric properties given in Lupishko
et al. (1995); Mukai et al. (1997); Ishiguro et al. (1997), we discuss the
physical properties of the surface.

For future reference, Table 1 lists the some important symbols we used in
Sections 2, 3, and Appendix A, their meanings, and their values and units.
Most source codes of the software we used for the data reduction and analysis
are publicly available (see Appendix B).

Table 1: Symbols frequently used in this paper

Category |  Symbols |  Description |  Value and Unit
---|---|---|---
Magnitudes |  $V_{\odot}$ |  Visual magnitude of the Sun |  $-26.762\mathrm{\,\mathrm{(mag)}}$
|  $V$, $\Delta V$ |  Visual magnitude and its uncertainty |  $\mathrm{\,\mathrm{(mag)}}$
|  $H_{\mathrm{V}}(\alpha)$ |  Reduced magnitude |  $\mathrm{\,\mathrm{(mag)}}$
|  $H_{\mathrm{V}}$ |  Absolute magnitude ($\equiv H_{\mathrm{V}}(0)$) |  $\mathrm{\,\mathrm{(mag)}}$
|  $G$ |  Slope parameter |  —
Ephemerides |  $r_{\mathrm{h}}$, $r_{\mathrm{g}}$ |  Heliocentric/geocentric distance |  $\mathrm{\,\mathrm{m}}$ or $\mathrm{\,\mathrm{au}}$
|  $\alpha$ |  Phase angle |  ($\mathrm{\,\mathrm{{}^{\circ}}}$ or $\mathrm{\,\mathrm{rad}}$)
|  $t$ |  Time on the target (light-time corrected) |  $\mathrm{\,\mathrm{s}}$ or $\mathrm{\,\mathrm{JD}}$
|  $(\lambda,\,\beta)$ |  Ecliptic coordinate (longitude, latitude) |  ($\mathrm{\,\mathrm{{}^{\circ}}}$ or $\mathrm{\,\mathrm{rad}}$)
Physical |  $S_{\mathrm{proj}}$ |  Total projected area viewed at $\alpha=0$ |  $\mathrm{\,\mathrm{m^{2}}}$
parameters |  $D$ |  Effective diameter |  $\mathrm{\,\mathrm{m}}$ or $\mathrm{\,\mathrm{km}}$
|  $p_{\mathrm{V}}$ |  Geometric albedo in visual (V) band |  —
|  $A_{5}$ |  Albedo at the phase angle of $\alpha=5^{\circ}$ |  —
|  $I$ |  The irradiance of the object of interest |  $\mathrm{\,\mathrm{W/m^{2}}}$
|  $F$ |  $I$ of a Lambertian reflector at normal incidence |  $\mathrm{\,\mathrm{W/m^{2}}}$
|  $I/F$ |  The radiance factor |  —

Parameters with “—” in the last column are dimensionless. The units are
usually in SI format unless special units are dominantly used in this work.
The units given in parentheses are dimensionless but are preferred to be
written out explicitly.

##  2 Observation and Data Reduction

Our observation journal is provided in Table 2. We used three 1.6-m telescopes
in the southern hemisphere that together comprise the Korea Microlensing
Telescope Network (or KMTNet, Kim et al. 2016) within the frame of the
DEEP–South project (Moon et al., 2016). Each telescope provides a
$2^{\circ}\times 2^{\circ}$ field of view via identical mosaic CCD cameras
with a pixel size of $0.4^{\prime\prime}$. The telescopes are located on three
continents: at the Cerro Tololo Inter–American Observatory (KMTNet-CTIO) in
Chile, the South African Astronomical Observatory (KMTNet-SAAO) in South
Africa, and the Siding Spring Observatory (KMTNet-SSO) in Australia. We
employed the standard Johnson–Cousins $V$-band filter system. We obtained
$\sim$ 10 frames per night at each observatory with individual exposure times
of 1 minute. The images were preprocessed by a reduction pipeline for KMTNet
data at the KMTNet data center, which performs bias subtraction, flat-
fielding, and cross-talk correction. A summary of the data reduction process
is given below (see also Figure 1).

Table 2: Summary of our observations

Date(a) | Time in UT(a) | Site(b) |  $N$(c) |  $r_{h}$(d) |  $r_{\mathrm{g}}$(d) |  $\alpha$(d) | $V$ | $\Delta V$ | $H_{\mathrm{V}}(\alpha)$ | Seeing(e) | Weather(e)
---|---|---|---|---|---|---|---|---|---|---|---
YYYY-MM-DD | HH:MM:SS |  |  | $\mathrm{\,\mathrm{au}}$ | $\mathrm{\,\mathrm{au}}$ | $\mathrm{\,\mathrm{{}^{\circ}}}$ | $\mathrm{\,\mathrm{mag}}$ | $\mathrm{\,\mathrm{mag}}$ | $\mathrm{\,\mathrm{mag}}$ | ′′ |
2018-04-07 | 03:12:30 | CTIO | 7 | 3.8067 | 2.8184 | 2.7863 | 21.503 | 0.172 | 16.350 | 1.0–1.8 | 0–1
2018-04-07 | 12:40:57 | SSO | 13 | 3.8078 | 2.8183 | 2.6618 | 21.231 | 0.105 | 16.078 | 1.3–1.4 | 0
2018-04-07 | 21:43:13 | SAAO | 10 | 3.8088 | 2.8182 | 2.5429 | 20.916 | 0.089 | 15.762 | 1.4 | 0
2018-04-09 | 03:10:59 | CTIO | 9 | 3.8122 | 2.8182 | 2.155 | 20.794 | 0.092 | 15.638 | 1.5–1.8 | 0
2018-04-09 | 12:38:01 | SSO | 10 | 3.8132 | 2.8183 | 2.0303 | 20.633 | 0.099 | 15.477 | 1.2–1.4 | 0
2018-04-11 | 12:45:45 | SSO | 5 | 3.8187 | 2.8196 | 1.3952 | 20.490 | 0.086 | 15.330 | 1.3–1.5 | 0
2018-04-11 | 21:43:31 | SAAO | 11 | 3.8197 | 2.8200 | 1.277 | 20.432 | 0.103 | 15.271 | 1.8 | 0
2018-04-13 | 03:09:13 | CTIO | 4 | 3.8230 | 2.8216 | 0.8906 | 20.486 | 0.124 | 15.322 | 1.2–1.6 | 0–1
2018-04-13 | 21:44:11 | SAAO | 13 | 3.8250 | 2.8228 | 0.649 | 20.770 | 0.091 | 15.603 | 1.7 | 0

(a)Average values of the UT date and time at the target (i.e., corrected for
the light travel time) of the images taken at the same night and the same
observatory. The total observing time in each night was less than 1 hour,
which is substantially smaller than the rotational and precessional periods of
Toutatis ($\sim 5$–$7\mathrm{\,\mathrm{days}}$).
(b)Telescope sites: Cerro Tololo Inter–American Observatory (CTIO), South
African Astronomical Observatory (SAAO), and Siding Spring Observatory (SSO).
(c)Number of exposures.
(d)Averaged over the $N$ ephemerides in a given night.
(e)Seeing and weather conditions from the official daily report (see Notebook
6 of Appendix B). Weather codes 0 and 1 correspond to “clear” and “thin
clouds”, respectively.

Figure 1: Summary of the data reduction process performed by teams at the
Korea Astronomy and Space Science Institute (KASI) and Seoul National
University (SNU). Photometry was conducted with publicly available open-source
packages indicated at the bottom left (the shadowed box), which includes
astroquery (Ginsburg et al., 2018), astroscrappy, ccdproc (Craig et al.,
2017), reproject (Robitaille, 2018), and photutils (Bradley et al., 2017), in
addition to the astropy core package (Astropy Collaboration et al., 2013,
2018).

The locations of Toutatis were specified using the Moving Object Detection
Program (MODP) developed by KASI. MODP updates the World Coordinate System
(WCS) information in the header of each image first and specifies the position
of the asteroid in each image. For convenience, we analyzed images clipped to
$2000\times 2000$ pixels ($800^{\prime\prime}\times 800^{\prime\prime}$)
centered on the Toutatis locations calculated by the MODP. Cosmic-ray signals
were removed with astroscrappy111https://github.com/astropy/astroscrappy
version 1.0.5 using a separable median filter. which uses the L. A. Cosmic
algorithm (van Dokkum 2001). We coadded all images taken in the same night at
the same observatory while shifting the individual images to increase the
signal-to-noise (S/N) ratio. Eventually, two composite images per night per
observatory were generated: one co-added image with sidereal offset and one by
centering the individual images on Toutatis.

The signals from sources detected in each composite image are compared with
those of field stars in the Pan-STARRS Data Release 1 (PS1, Flewelling et al.
2016). We first cone-searched PS1 objects within $0.06^{\circ}$ centered at
Toutatis and with more than 5 observations in the $g$ and $r$ bands. We then
used the criterion $\texttt{iMeanPSFMag}-\texttt{iMeanKronMag}>0.05$ to reject
non-stellar
objects,222https://outerspace.stsci.edu/display/PANSTARRS/How+to+separate+stars+and+galaxies
where iMeanPSFMag and iMeanKronMag are the mean i-band magnitudes obtained
from using point spread function and Kron radius photometry, respectively. If
saturated stars resulted in blooming and/or contaminated nearby pixels, all
affected pixels were manually masked, and stars located closer than $\sim 6d$,
where $d$ is the full-width at half-maximum of the seeing disk, were rejected.
We fixed $d$ to be 4 pixels ($\sim 1.6^{\prime\prime}$). In addition, nearly
colocated stars were identified and rejected by using the DAOGROUP algorithm
(Stetson 1987 implemented via photutils by Bradley et al. 2017) with a minimum
separation distance of $7.5d$.

Next, we used the transformation formula

$V=g-0.59(g-r)-0.01$ (1)

from Lupton
(2005)333https://www.sdss.org/dr14/algorithms/sdssUBVRITransform/#Lupton2005
to obtain the Johnson–Cousins $V$-band magnitudes of the stars, where $g$ and
$r$ are the mean aperture photometry magnitudes (MeanApMag) of the $g$\- and
$r$-band, respectively. We fit a line to $V$ and the instrumental magnitude
from circular aperture photometry of the stars (aperture radius $1.5d$, inner
and outer sky radii $4d$ and $6d$, respectively) in the sidereally coadded
images; only stars with $15\leq V\leq 20$ are used for the analysis. The
regression line provides the amount of atmospheric extinction in relative
photometry, and we used it to obtain the V-band magnitude of Toutatis in the
Toutatis-centered images via extrapolation. The minimum number of stars used
in the regression (in the April 7 data from CTIO) was nine. Due to the
extrapolation, the uncertainty in $V$, $\Delta V$ (obtained from the
1-$\sigma$ confidence interval of the regression line), must be understood as
a lower limit of the true uncertainty.

After data reduction, we compared the results with the raw images and found an
artificial pattern contaminating one of the datasets (2018-04-11 SAAO). For
this special case, we did manual photometry (selecting appropriate sky regions
by visual inspection); all other data sets were analyzed as described above.
We summarize the observational quantities in Table 2 and plotted them as a
function of time in Figure 2.

##  3 Derivation of the Geometric Albedo

###  3.1 Correction for Rotation and Distances

The observed $V$ varies not only because of rotation and variation of the
solar phase angle ($\alpha$, the Sun–asteroid–observer angle) but also because
of changes in the heliocentric and observer distances ($r_{\mathrm{h}}$ and
$r_{\mathrm{g}}$, respectively). We first corrected the effects of the
heliocentric and observer distances by deriving the reduced magnitude
$H_{\mathrm{V}}(\alpha)$, which is defined as a magnitude when the asteroid is
viewed at unit heliocentric and observers distance (i.e.,
$r_{\mathrm{h}}=r_{\mathrm{g}}=1\mathrm{\,\mathrm{au}}$) but at arbitrary
phase angles. This parameter is given by

$H_{\mathrm{V}}(\alpha)=V(r_{\mathrm{h}},r_{\mathrm{g}},\alpha)-5\log_{10}\left(\frac{r_{\mathrm{h}}r_{\mathrm{g}}}{1\mathrm{\,\mathrm{au}}\cdot
1\mathrm{\,\mathrm{au}}}\right)~{}.$ (2)

The IAU $H$, $G$ magnitude system (Bowell et al., 1989) describes the
$\alpha$-dependence of $H_{\mathrm{V}}(\alpha)$ using two parameters, the
absolute magnitude $H_{\mathrm{V}}=H_{\mathrm{V}}(\alpha=0)$, and the slope
parameter $G$:

$H_{\mathrm{V}}(\alpha)=H_{\mathrm{V}}+2.5\log_{10}[(1-G)\Phi_{1}(\alpha)+G\Phi_{2}(\alpha)]~{},$
(3)

where

$\displaystyle\Phi_{1}(\alpha)$ $\displaystyle=$
$\displaystyle\exp\left[-3.33\tan^{0.63}(\alpha/2)\right]~{},$
$\displaystyle\Phi_{2}(\alpha)$ $\displaystyle=$
$\displaystyle\exp\left[-1.87\tan^{1.22}(\alpha/2)\right]~{}.$ (4)

Our observations were made at very small $\alpha$ values. For a spherical
object, the fraction of dark area (outside the terminator) is
$f_{\mathrm{dark}}=(1-\cos\alpha)/2$. Since our observations were made at
$\alpha<3^{\circ}$, $f_{\mathrm{dark}}<0.07\mathrm{\,\mathrm{\%}}$, and a non-
spherical shape, such as that of Toutatis, does not significantly increase
this value. Therefore, we assume that the total projected area of the target
viewed from a given direction will be the same as that of the sunlit surface.

Furthermore, the reflected intensity per projected area (which is closely
related to the radiance factor; see Section 3.2) is nearly independent of the
incidence and emission angles and depends only on the phase angle $\alpha$,
when the object is viewed near opposition. Thus, the flux reflected from a
surface patch on Toutatis is proportional to its projected area, regardless of
its orientation. This was first conjectured observationally by Markov &
Barabashev (1926) from lunar observations as mentioned in the appendix of
Dollfus & Bowell (1971). It was confirmed by Lee & Ishiguro (2018) using a
modern theoretical model: their Figure 4 shows that the radiance factor
(defined in Section 3.2) is reduced only by $\sim 1\mathrm{\,\mathrm{\%}}$ or
$\sim 3\mathrm{\,\mathrm{\%}}$ when the incidence angle is chagned from
$0^{\circ}$ to $70^{\circ}$ or $90^{\circ}$ (the azimuthal angle and phase
angle are fixed to 0). This means, if we correct the phase effect using
Equation (3), the obtained $H_{\mathrm{V}}$ will be linearly related to the
logarithm of the geometrical cross-sectional area (Equation 5 below).

Figure 2 shows $H_{\mathrm{V}}(\alpha)$ and $H_{\mathrm{V}}$ as function of
time, using $G=0.10$ following Spencer et al. (1995). The best rotational
model, based on the high-resolution radar shape model of Hudson et al. (2004),
is shown as red solid line, and other models within the 1-$\sigma$ confidence
interval are shown as faint black dotted lines. The process of selecting
rotational models is described in Appendix A. We confirmed that any $G$ with
$0.10\leq G\leq 0.50$ does not change our results. Our reduced magnitudes and
the amplitude of the light curve agree with previously reported results
(Spencer et al., 1995).

###  3.2 Radiance Factor and Geometric Albedo

The radiance factor is given by $I/F$ where $I$ is the measured irradiance of
the object, whereas $F$ is that of a perfectly diffuse (Bond albedo of unity)
Lambertian plate444Using a Lambertian plate rather than a Lambertian sphere
has historical rather than mathematical or physical reasons (see, e.g.,
Russell, 1916). If we used a Lambertian sphere, the $F$ value would be
$1/2\pi$ rather than $1/\pi$. of the same projected area at the identical
geometric configuration to that of the object, but oriented such that the
incidence angle of sunlight is $0$. Since the projected area is the same for
both $I$ and $F$, the ratio $I/F$ will be identical to the ratio of the
bidirectional reflectance of the object (which is unknown) and the Lambertian
plate (which is $\cos(0)/\pi=1/\pi$). Thus, at a perfect opposition
($\alpha=0^{\circ}$), $I/F$ coincides with $p_{\mathrm{V}}$ by definition.

Figure 2: Reduced magnitude, $H_{\mathrm{V}}(\alpha)$, and phase-corrected
magnitude, $H_{\mathrm{V}}$, as function of time. The numbers above each
marker indicate the phase angles $\alpha$ at the observation epoch. The red
solid line is the best-fit geometrical cross-sectional area,
$-2.5\log_{10}(S_{\mathrm{proj}}(t))$ (See Equation 5), the faint black dotted
lines indicate model curves within the 1-$\sigma$ confidence interval (See
Appendix A). The short-time ($\sim$1–2 days) bumpy features in the data are
reproduced in the model, meaning that these features are attributed to the
rotation of the asteroid.

As described in Section 3.1, the $I/F$ value remains almost constant over the
surface when the object is observed near opposition. Thus, the observed flux
$F_{\mathrm{obs}}$ is proportional to $I/F(\alpha)\times
S_{\mathrm{proj}}(t)$, where $I/F(\alpha)\approx\mathrm{const}$ over the
surface of Toutatis. The obtained reduced magnitude, $H_{\mathrm{V}}(\alpha)$,
can then be converted into a logarithm of $I/F$:

$-2.5\log_{10}\hskip-2.0pt\left(\frac{I}{F}\right)\hskip-2.0pt=\hskip-2.0ptH_{\mathrm{V}}(\alpha)-V_{\odot}-2.5\log_{10}\hskip-2.0pt\left(\frac{\pi}{S_{\mathrm{proj}}(t)}\right)+m_{c},$
(5)

where $V_{\odot}=-26.762$ is the V-band magnitude of the Sun at 1 au (Mann &
von Braun, 2015), $S_{\mathrm{proj}}(t)$ is the geometrical cross section in
$\mathrm{\,\mathrm{m^{2}}}$ as a function of time, and
$m_{c}=-5\log_{10}(1\mathrm{\,\mathrm{au}}/1\mathrm{\,\mathrm{m}})=-55.87$ is
a constant to adjust the length unit.

The calculated $I/F$ values are plotted against the phase angle $\alpha$ in
the upper panel of Figure 3. We used a single power-law
($I/F(\alpha)=p_{\mathrm{V}}\times 10^{b\alpha}$) to fit the points
approximately. From this plot, we considered four parameters of interest: the
geometric albedo ($p_{\mathrm{V}}$), the albedo at phase angle of $5^{\circ}$
($A_{5}$), and two factors

$f_{1}=\frac{I(0.3^{\circ})}{I(5^{\circ})}\quad;\quad
f_{2}=\frac{I/F(0.7^{\circ})-I/F(2.5^{\circ})}{2.5^{\circ}-0.7^{\circ}}~{},$
(6)

which were used to check the validity of our results by comparing with
previous works (explained below).

Figure 3: _Top:_ The radiance factor ($I/F$) as a function of phase angle
($\alpha$). The observed radiance factor values (blue cross markers) are
fitted using a single power-law: $I/F(\alpha)=p_{\mathrm{V}}\times
10^{b\alpha}$. The best fit $p_{\mathrm{V}}$, $f_{1}$, and $f_{2}$ (see
Equation 9) and their 1-$\sigma$ uncertainties are shown above the figure. The
red solid line is the best-fit model, the shaded area marks the 1-$\sigma$
confidence interval of the model fit. The black marker shows
$A_{5}=I/F(\alpha=5^{\circ})$ and its 1-$\sigma$ confidence interval,
indicated by the numbers next to the marker. _Bottom:_ The corresponding
$\chi^{2}_{\mathrm{red}}$ statistic in the parameter space. The 1-$\sigma$
confidence contour of $\chi^{2}_{\mathrm{red}}$ and that of $p_{\mathrm{V}}$
are shown. The colormap is in log-scale and contours for a few selected
$\chi^{2}_{\mathrm{red}}$ values are shown to guide the eyes.

We used a brute-force grid search algorithm (using a fixed grid) to find the
best-fit power-law parameters $p_{\mathrm{V}}$ and $b$. The best-fit
parameters were first found by simple $\chi^{2}$ minimization where the
reduced $\chi^{2}$ is defined as

$\chi^{2}_{\mathrm{red}}:=\frac{1}{N_{\mathrm{obs}}-2}\sum_{i=1}^{N_{\mathrm{obs}}}\left(\frac{I/F(\alpha_{i})-p_{\mathrm{V}}\times
10^{b\alpha_{i}}}{\sigma_{i}}\right)^{2}.$ (7)

Here, $N_{\mathrm{obs}}=9$ is the total number of data points,
$N_{\mathrm{obs}}-2=7$ is the number of degrees of freedom, and $\sigma_{i}$
is the uncertainty in $I/F$ for the $i$-th observation. The
$\chi^{2}_{\mathrm{red}}$ contours in the 2-D parameter space (bottom panel of
Figure 3) are carefully investigated to assure that we searched a large enough
parameter space with a sufficiently fine grid. The uncertainties of $f_{1}$,
$f_{2}$, and $A_{5}$ are estimated based on the minimum and maximum values of
each of them for the models within the 1-$\sigma$ confidence interval (the
gray region in the top panel of Figure 3).

From the fit, the amplitude parameter is the geometric albedo by definition
($p_{\mathrm{V}}\equiv I/F(0)$). We obtained the geometric albedo in the
V-band $p_{\mathrm{V}}=0.185^{+0.045}_{-0.039}$. This is a typical albedo
value for an S-type asteroid.

The albedo at $\alpha=5^{\circ}$, $A_{5}$, is important since this is used in
experimental studies to compare with celestial bodies (see Section 4.4). It is
defined as the ratio of the measured flux of the obeject of interest at
$\alpha=5^{\circ}$ to that of a perfectly diffuse Lambertian plate at the same
geometrical configuration with the same projected area.555The angle of
$\alpha=5^{\circ}$ is chosen by experimentalists, since the albedo
measurements were found to give the most reproducible results with
$\alpha=5^{\circ}$ (see Geake & Dollfus, 1986, and references therein). A
smoked MgO screen illuminated at an incidence angle of $0^{\circ}$ is used as
an analog of Lambertian plate. The $A_{5}$ is the ratio of the flux of the
target of interest and the MgO screen, while both fluxes are measured at
$\alpha=5^{\circ}$ with zero incidence angle. Thus,

$A_{5}=\frac{I}{F}(\alpha=5^{\circ})=0.104^{+0.050}_{-0.034}~{},$ (8)

as the black marker shown in Figure 3. The two factors, $f_{1}$ and $f_{2}$
are

$f_{1}=1.708^{+1.285}_{-0.740}\quad;\quad
f_{2}=0.018^{+0.019}_{-0.019}\mathrm{\,\mathrm{/^{\circ}}}~{}.$ (9)

$f_{1}$ was investigated in Belskaya & Shevchenko (2000) and found to be
$f_{1}=1.34\pm 0.05$ for C-type and $f_{1}=1.44\pm 0.04$ for S-type asteroids.
From the V-filter result in Figure 7 of Tatsumi et al. (2018), $f_{1}\sim 1.3$
for the asteroid (25143) Itokawa, an S-type asteroid. $f_{2}$ was determined
for asteroid Itokawa in Lee & Ishiguro (2018). For Itokawa, they found
$f_{2}\sim 0.01\pm 0.01$ for $\alpha>1.4^{\circ}$ and $f_{2}\sim 0.04\pm 0.01$
for $\alpha<1.4^{\circ}$ (note that the sign convention is opposite from the
one for their $s_{\mathrm{OE}}$). We employed $0.7^{\circ}$ and $2.5^{\circ}$
because the phase angle difference is larger than $0.3^{\circ}$ as Lee &
Ishiguro (2018) and this is the interpolated region. Both $f_{1}$ and $f_{2}$
coincide with the expected values from previous works, but it is difficult to
arrive at further conclusions due to the uncertainties.

Our parameter values are summarized in Table 3. In the last column of the
table, we summarized the values of the same parameters but derived by
excluding one of the data points (and updating the number of degrees of
freedom accordingly), 2018-04-07 CTIO with $\alpha=2.79^{\circ}$, since this
is the observation with the largest deviation from the shape model prediction
(Figure 2). Comparing the two results, it can be seen that the derived
parameters are robust against the exclusion of the deviant data point
(parameters changed much less than the 1-$\sigma$ confidence range).

Table 3: Parameter values derived using either all observational data or after excluding the data of 2018-04-07 CTIO ($\alpha=2.79^{\circ}$) Parameters | All data | Exclude $\alpha=2.79^{\circ}$
---|---|---
$p_{\mathrm{V}}$ | $0.185^{+0.045}_{-0.039}$ | $0.179^{+0.042}_{-0.037}$
$A_{5}$ | $0.104^{+0.050}_{-0.034}$ | $0.116^{+0.055}_{-0.039}$
$f_{1}$ | $1.708^{+1.285}_{-0.740}$ | $1.509^{+1.118}_{-0.649}$
$f_{2}\mathrm{\,\mathrm{[/^{\circ}]}}$ | $0.018^{+0.019}_{-0.019}$ | $0.014^{+0.019}_{-0.019}$

##  4 Discussion

The geometric albedos of asteroids have been derived with several techniques,
including (1) infrared observations of thermal radiation, (2) time-delay and
Doppler measurements of radar signals emitted from ground-based stations, (3)
measurements of the time duration during occultation of background stars, (4)
direct imaging with adaptive optics techniques, (5) in situ measurements of
the bidirectional reflectance using spacecraft cameras, and (6) an empirical
method using polarimetric slopes. Since techniques (1)–(4) are contrived to
derive the sizes (and shapes) rather than the albedo values of asteroids, it
is essential to combine the visible magnitudes to derive the albedos using
these four techniques. It is, however, true that optical observations have not
been conducted strenuously despite recent comprehensive surveys, especially by
means of (1) and (2). In general, the visible magnitudes have been obtained
using uncalibrated observation data on the presumption that asteroids have a
phase function (typically $G=0.15$ for an IAU $H$, $G$ magnitude system). In
fact, it is quite difficult to make optical observations of asteroids at the
opposition point (or at least within a hypothetical circle having the equal
apparent diameter of the Sun) because the orbital planes of almost all
asteroids do not coincide with Earth’s orbital plane (i.e., $i\neq 0$). Fast
time variations of the cross-sectional areas of asteroids also make it
difficult to derive the albedo values even when the phase function and
dimension of the asteroids are known. This effect is especially significant
for elongated asteroids. In contrast, direct observations of asteroidal
surfaces via space missions provide more reliable information about the
albedos, eliminating the above uncertainties regarding the cross-sectional
areas. However, note that the calibration processes of onboard data have some
uncertainties, such as the standardization of magnitude systems and
unavailability of flat-fielding data in space. Some in situ data were acquired
at large phase angles (e.g., (433) Eros by the NEAR-Shoemaker mission). Thus,
it is not straightforward to derive the accurate geometric albedo values.

###  4.1 Sources of Uncertainties

The parameters we obatained in this study, especially the albedos, may have
suffered from errors that are difficult to quantify correctly. For one of our
observations (2018-04-07 CTIO with $\alpha=2.79^{\circ}$), the magnitude is
$\sim 1$-$\sigma$ fainter than the model prediction (Figure 2). Although a
1-$\sigma$ of deviation is not necessarily significant, there is a possibility
that the value actually was affected by an unknown systematic shift, such as
those caused by imperfect phase function modeling (Equation 3) and weather
effects (Table 2). Although it was almost invisible, some artifact as in
2018-04-11 SAAO (see the last part of Section 2) may have affected this image
in a non-trivial way. This data, however, does not seem to critically affect
the quantities derived in this work, as shown in Table 3.

In addition, as the radiance factor calculation requires the absolute physical
value of the projected area (Equation 5), the uncertainty in the physical size
affects the $I/F$ value. From Equation (5), $I/F\propto 1/S_{\mathrm{proj}}$
($\sim D^{-2}$ to first order), if all other quantities including the observed
values are kept constant. The radar models we used give the maximum sizes
along each principal axes as $(4.60,\,2.28,\,1.92)\pm
0.10\mathrm{\,\mathrm{km}}$ (Hudson et al., 2003), which is slightly smaller
but within the uncertainty ranges obtained by Chang’e-2, $(4.75\pm
0.48,\,1.95\pm 0.19)\mathrm{\,\mathrm{km}}$ (Huang et al., 2013). Ignoring the
uncertainty ranges, $S_{\mathrm{proj}}$ may have been underestimated
systematically by $\sim 5\mathrm{\,\mathrm{\%}}$ in our estimation. Hence, our
albedo estimations, $p_{\mathrm{V}}\equiv I/F(\alpha=0)$ and $A_{5}\equiv
I/F(\alpha=5^{\circ})$, could have been overestimated by $0.009$ and $0.005$,
respectively. These uncertainties are negligible compared to the statistical
uncertainties (only $\sim 10\mathrm{-}25\mathrm{\,\mathrm{\%}}$ of the error-
bars; see Table 3).

###  4.2 Comparison with Flyby Observations

The Chinese Chang’e-2 spacecraft succesfully obtained optical images of
Toutatis during its December 2012 flyby at a distance of $\sim
1.3\mathrm{\,\mathrm{km}}$, and the hemispherical albedos were derived (Zhao
et al., 2016). These results allow testing of the albedo estimates gained from
different approaches including this work. It is, however, difficult to
directly compare their results to ours since they estimated the hemispherical
albedo (assumed to be equal to the Bond albedo, $A_{\mathrm{B}}$), different
from bidirectional albedos derived in this work (e.g., $p_{\mathrm{V}}$ and
$A_{5}$).

Using the IAU H, G system (Bowell et al., 1989), the Bond albedo can be
obtained from $p_{\mathrm{V}}$ using a numerical approximation of the phase
integral:

$A_{\mathrm{B,V}}=(0.286+0.656G)p_{\mathrm{V}}~{},$ (10)

where $A_{\mathrm{B,V}}$ is the Bond albedo in V-band.666Conventionally,
$A_{\mathrm{B,V}}=(0.290+0.684G)p_{\mathrm{V}}$ suggested by Bowell et al.
(1989) is widely being used. However, Myhrvold (2016) found that the best
fitting function is $A_{\mathrm{B,V}}=(0.286+0.656G)p_{\mathrm{V}}$. We
confirmed the latter result and used it here. Using the uncertainty in
$p_{\mathrm{V}}$ and $G=0.0\mathrm{-}0.2$, we obtain extrema of
$A_{\mathrm{B,V}}=0.04\mathrm{-}0.10$. This is smaller than the values derived
in Zhao et al. (2016): 0.2083, 0.1269, and 0.1346 in the R, G, and B bands of
Chang’e-2, respectively. Using Equation (10), these can be transformed into
the geometric albedos of $0.50\mathrm{-}0.73$, $0.30\mathrm{-}0.44$, and
$0.32\mathrm{-}0.47$ assuming $G=0.0\mathrm{-}0.2$ and identical phase
functions, respectively.

The discrepancy may have been caused by the assumptions used to compare the
two physically different albedos (bidirectional and Bond). To measure the Bond
albedo directly, one needs to observe the object at all the possible
$4\pi\mathrm{\,\mathrm{sr}}$ directions, which is impractical. Thus, we
assumed a phase function (Equation 10) to convert the $p_{\mathrm{V}}$ to
$A_{\mathrm{B,V}}$, while Zhao et al. (2016) assumed illumination conditions
(incidence angles) to convert the radiance ($\mathrm{W\,m^{-2}\,sr^{-1}\,\mu
m^{-1}}$) into the albedo, for example. The phase angle coverages of the
observations in both works are also different.

Therefore, this discrepancy is a result of imperfect assumptions used to
indirectly infer the Bond albedo in the two works, while these systematic
uncertainties are not taken into account in the error bars. We emphasize that
our $p_{\mathrm{V}}$ and $A_{5}$ should be accurate within the derived
uncertainty range, because these were obtained from observations made at
$\alpha=0.6^{\circ}\mathrm{-}2.8^{\circ}$, close to where these quantities are
defined.

###  4.3 Comparison with Thermal Infrared Studies

The sizes and albedos provided by infrared catalogs such as IRAS (Tedesco et
al., 2002), AKARI (Usui et al., 2011), and NEOWISE (Masiero et al., 2017)
provide an invaluable overview on asteroids of different taxonomic types
distributed across the solar system; nonetheless, it is also important to note
that some asteroids with a small number of detections and elongated shapes
have large uncertainties due to rotation.

Masiero et al. (2017)777The machine-readable table is available at
http://iopscience.iop.org/1538-3881/154/4/168/suppdata/ajaa89ect1_mrt.txt
obtained $\log_{10}p_{\mathrm{V}}=-0.392\pm 0.166$ with a diameter $D=1.788\pm
0.376\mathrm{\,\mathrm{km}}$ for Toutatis from 6 NEOWISE infrared observations
in both W1 and W2 bands at phase angle $\alpha\sim 48^{\circ}$, in combination
with $H=15.30$, $G=0.10$, and a beaming parameter $\eta=1.007\pm 0.223$. The
$p_{\mathrm{V}}$ value can be transformed to
$p_{\mathrm{V}}=0.406^{+0.189}_{-0.129}$, if we assume the posterior
distribution of their $\log_{10}p_{\mathrm{V}}$ estimation is truly, or at
least similar to, Gaussian. It should be noted that the observations were made
at a single phase angle, so the Near Earth Asteroid Thermal Model (NEATM) used
in Masiero et al. (2017) may not give an accurate estimation, but other
advanced modelings may be more adequate, such as the advanced thermophysical
model (ATPM; Rozitis & Green 2011, which is used for space mission related
targets, e.g., Yu et al. 2014, 2015).

Meanwhile, Usui et al. (2014) noted a 22% deviation for the 1-$\sigma$
confidence range among three radiometric survey catalogs by IRAS, AKARI, and
WISE (Tedesco et al., 2002; Usui et al., 2011; Mainzer et al., 2011). The
deviation is caused not only by the different types of thermal models but also
by the observed rotational phases of the asteroids. It is worth noting that
the size and albedo determination can be more uncertain for individual targets
than the estimated 22% from the average trend.

Note also the currently ongoing debate about the accuracy of the NEOWISE
results (see, e.g., Myhrvold, 2018a; Wright et al., 2018; Myhrvold, 2018c).
According to Wright et al. (2018), however, the results we used for Toutatis
(Masiero et al., 2017) are not affected by any problem. Meanwhile, Myhrvold
(2018b) discusses a new approach to the NEATM by introducing a new
parametrization and a physically valid yet simple generalization for the case
that the reflected sunlight is not negligible in the thermal spectra. This new
approach may improve the NEOWISE results with respect to accuracy and
consistency.

In case of Toutatis, one needs to pay close attention to the rotational phase
because of the very elongated shape of the asteroid. In fact, the size given
in the NEOWISE catalog is only 73 % of the mean diameter derived from radar
observations (i.e., $D=2.45\mathrm{\,\mathrm{km}}$). Using the relationship
$D=1329\mathrm{\,\mathrm{[km]}}p_{\mathrm{V}}^{-0.5}10^{-H_{\mathrm{V}}/5}$
(derived in, e.g., the appendix of Pravec & Harris, 2007), the
$p_{\mathrm{V}}$ value appears to be overestimated by a factor of $\sim
1/0.73^{2}=1.88$. Despite this not being an accurate statement since the
change in $p_{\mathrm{V}}$ changes the shape of the thermal spectrum of the
asteroid, we may crudely estimate the updated albedo to be $p_{\mathrm{V}}\sim
0.406/1.88=0.22$. This coincides with the albedo expected for an S-type
asteroid.

###  4.4 Comparison with Polarimetric Studies

Lupishko et al. (1995) studied the polarimetric properties of Toutatis and
derived $p_{\mathrm{V}}=0.13$ using the empirical polarimetric slope–albedo
law stated in Zellner et al. (1977). Subsequently, Mukai et al. (1997) derived
the same value, $p_{\mathrm{V}}=0.13$, using the polarimetric slope–albedo law
in Dollfus & Zellner (1979). The empirical relationship, first noted in Widorn
(1967), is

$\log_{10}p_{\mathrm{V}}=C_{1}\log_{10}h+C_{2}~{},$ (11)

where $h$ denotes the polarimetric slope around the inversion angle in
$\mathrm{\,\mathrm{\%/^{\circ}}}$ and where $C_{1}$ and $C_{2}$ are constants.
Note that the albedo in Zellner et al. (1977) and Dollfus & Zellner (1979) is
different from the one that we currently apply. Thus, these authors derived a
set of constants $(C_{1},\,C_{2})=(-0.93,\,-1.78)$ based on the laboratory
polarimetric measurements, where the measurements defined the albedo at the
normal ($0^{\circ}$) incidence and $5^{\circ}$ angle of emergence (i.e.,
$\alpha=5^{\circ}$, not $0^{\circ}$). It is challenging to measure the
geometric albedo, which is defined at $\alpha=0^{\circ}$, because the light
source is in the line of sight of the detector, thus hiding the sample.
Although we acknowledge these early efforts, the polarimetric albedo in
Lupishko et al. (1995) and Mukai et al. (1997) needs to be updated to match
the definition of the geometric albedo.

Multiple studies made efforts to derive the set of constants for obtaining the
geometric albedo from the polarimetric slope (Masiero et al., 2012; Cellino et
al., 2015; Lupishko, 2018), involving better ways to calibrate the data from
different sources. We estimated the geometric albedos using the polarimetric
slope–albedo law using these sets of parameters, and the results are
summarized in Table 4. To first order, the Gaussian 1-$\sigma$ confidence
range of $p_{\mathrm{V}}$ is estimated from Equation (11) and

$\Delta p_{\mathrm{V}}\sim\sqrt{\left(\frac{\partial p_{\mathrm{V}}}{\partial
C_{1}}\right)^{2}\hskip-5.0pt\Delta C_{1}^{2}+\left(\frac{\partial
p_{\mathrm{V}}}{\partial C_{2}}\right)^{2}\hskip-5.0pt\Delta
C_{2}^{2}+\left(\frac{\partial p_{\mathrm{V}}}{\partial
h}\right)^{2}\hskip-5.0pt\Delta h^{2}}~{},$ (12)

where $\Delta X$ is the Gaussian 1-$\sigma$ confidence range for the parameter
$X$. We summarize the geometric albedo derived using different observations
and different sets of parameters in Table 4. These polarimetric results are
consistent with the geometric albedo that we derived through the direct
photometric observation at opposition.

Table 4: Geometric albedo ($p_{\mathrm{V}}$) values obtained from polarimetry

Reference | $C_{1}$ | $C_{2}$ | Lupishko et al. (1995) | Mukai et al. (1997)
---|---|---|---|---
Masiero et al. 2012 | $-1.207\pm 0.067$ | $-1.892\pm 0.141$ | $0.182\pm 0.066$ | $0.186\pm 0.073$
Cellino et al. 2015 | $-1.111\pm 0.031$ | $-1.781\pm 0.025$ | $0.190\pm 0.022$ | $0.194\pm 0.034$
Lupishko 2018 | $-0.989\pm 0.047$ | $-1.719\pm 0.040$ | $0.168\pm 0.025$ | $0.171\pm 0.033$

$C_{1}$ and $C_{2}$ are constants in the polarimetric slope–albedo law given
by $\log_{10}p_{\mathrm{V}}=C_{1}\log_{10}h+C_{2}$. These parameters are
determined by the workscited in the left column. We used the slope parameters
given in Lupishko et al. (1995) and Mukai et al. (1997).

Finally, we derived the grain size on the surface. The empirical relationships
regarding the grain sizes (Geake & Dollfus, 1986) include $A_{5}$, not
$p_{\mathrm{V}}$. In Section 3.2, we derived $A_{5}=0.104^{+0.050}_{-0.034}$
in the V-band. From previous polarimetric data, it has been found that
$A_{5}=0.13\pm 0.02$ via the slope–albedo law (Lupishko et al., 1995; Mukai et
al., 1997). By comparing with terrestrial rock powders (Figure 4), the grain
size is estimated as $d\lesssim 50\mathrm{\,\mathrm{\mu m}}$.

The grain size is smaller than that of (3200) Phaethon ($d\gtrsim
360\mathrm{\,\mathrm{\mu m}}$, Ito et al. 2018) and (1566) Icarus ($d\sim
100-130\mathrm{\,\mathrm{\mu m}}$, Ishiguro et al. 2017) that were calculated
by using the same relationship. In addition, we compare the
$P_{\mathrm{max}}$–$A_{5}$ relationship of these three asteroids and
laboratory sample data (Figure 4). It is likely that (3200) Phaethon and
(1566) Icarus are covered with grains larger than that of Toutatis, although
other factors such as the porosity of grains can affect the size estimate.
These results may suggest that the near-Sun environment at less than
approximately $0.2\mathrm{\,\mathrm{au}}$ affects the paucity of small grains
on the surface.

Note that the observations or calculations of $A_{5}$ and $P_{\mathrm{max}}$
for these asteroids were sometimes made at wavelengths different from those
used in the laboratory experiments in Geake & Dollfus (1986). Nevertheless,
the deviations that can be expected are within the measurement uncertainties.

Figure 4: Albedo at $\alpha=5^{\circ}$ vs. $P_{\mathrm{max}}$ plot for lunar
and terrestrial samples in Geake & Dollfus (1986) along with those of three
asteroids whose $P_{\mathrm{max}}$ values are well studied. The data for
(3200) Phaethon and (1566) Icarus are from Ishiguro et al. (2017) and Ito et
al. (2018), respectively. The word “Terr” in the legend means “terrestrial”
materials. The numbers near the three asteroidal objects denote the perihelion
distances of each asteroids in au.

##  5 Conclusions

In this work, we analyzed imaging data of the $\mathrm{S_{k}}$-type asteroid
(4179) Toutatis at near-opposition configuration obtained by KMTNet. We
obtained a robust geometric albedo value of
$p_{\mathrm{V}}=0.185^{+0.045}_{-0.039}$ from the observed radiance factors
and the shape model. Our value is typical for an S-complex asteroid, and thus
shows the future potential of using KMTNet for the opposition monitoring of
asteroids. Other measures of albedo using the radiance factors ($f_{1}$ and
$f_{2}$), which were introduced in previous works, agree with our results
within the uncertainties albeit it is substantial. Thermal modeling and
polarimetry give consistent results if we use appropriate diameter values or
empirical relationships, respectively. The albedo at $\alpha=5^{\circ}$ is
also determined from the radiance factor and is consistent with Toutatis being
covered with particles smaller than the asteroids in the near-Sun environment
found from empirical relationships involving the grain size.

###### Acknowledgements.

We thank the two anonymous reviewers for their valuable comments which led to
important improvements of the manuscript. This research has made use of the
KMTNet system operated by the Korea Astronomy and Space Science Institute
(KASI) and the data were obtained at three host sites of CTIO in Chile, SAAO
in South Africa, and SSO in Australia. The observations around the opposition
were made using telescopes in the frame of contract research between Seoul
National University (SNU) and the Korea Astronomy and Space Science Institute
(KASI) for the DEEP–South project. This research was supported by the Korea
Astronomy and Space Science Institute under the R&D program supervised by the
Ministry of Science, ICT and Future Planning.

##  Appendix A Rotational Model

We calculated the projected geometrical cross-sectional area,
$S_{\mathrm{proj}}(t;\varphi_{1},\varphi_{2})$ of the asteroid as seen from
Earth at the epoch of the observations ($t$), assuming arbitrary rotational
and precessional offsets, $\varphi_{1}$ and $\varphi_{2}$. In the calculation,
we employed the high-resolution shape
model888https://sbn.psi.edu/pds/resource/rshape.html (Hudson et al., 2003,
2004). We considered the long-axis mode rotation and precession along
$(\lambda,\,\beta)=(180^{\circ},\,-52^{\circ})$ in the ecliptic coordinate
(Hudson & Ostro, 1995). Although these rotation and precession periods have
been reported to be $P_{1}=5.38\mathrm{\,\mathrm{days}}$ and
$P_{2}=7.40\mathrm{\,\mathrm{days}}$, respectively (Zhao et al., 2015), it is
necessary to perform fine tuning by adding the offsets $\varphi_{1}$ and
$\varphi_{2}$ to fit the observed magnitudes to the modeled ones because the
periods in the model are not determined well enough to be extrapolated to the
epoch of our observations (uncertainty in periods multiplied by the time
difference is much larger than the periods themselves). We thus defined the
rotational angle along the long-axis and the precession axis ($\theta_{1}$ and
$\theta_{2}$, respectively) as follows:

$\theta_{j}(t,\,\varphi_{j})=\frac{2\pi(t-t_{0})}{P_{j}}+\varphi_{j}~{},$ (13)

for $j=1,\,2$, where $t_{0}$ denotes the time of the reference epoch.

We used IDL POLYSHADE to calculate the total projected area
$S_{\mathrm{proj}}(t;\varphi_{1},\varphi_{2})$ for a given pair of
$(\varphi_{1},\,\varphi_{2})$ with grid size of $\Delta\varphi_{1}=5^{\circ}$
and $\Delta\varphi_{2}=2^{\circ}$ and time step size of $1/3$ days. The
projected area as a function of the direction to the observer is shown in
Figure 5. The color and contour show the projected area viewed by an observer
at an infinite distance to the direction specified the body-fixed frame,
$(\theta_{\mathrm{bf}},\,\phi_{\mathrm{bf}})$. As can be seen, an uncertainty
of $\sim 5^{\circ}$ in the viewing direction causes a fractional uncertainty
(normally $\ll 10\mathrm{\,\mathrm{\%}}$) which is much smaller than the
photometric uncertainty of $H_{\mathrm{V}}(\alpha)$ in Figure 2 and Table 2.
Thus, the choices for the grid size of the offsets ($\Delta\varphi_{1}$ and
$\Delta\varphi_{2}$) are reasonable.

Each of the 12,960 models for $(\varphi_{1},\,\varphi_{2})$ pair is then
spline interpolated when necessary. As mentioned in Section 3.1, the total
reflected flux is proportional to the projected area when $\alpha$ is small.
Hence, for each rotational model, we fit $a$ such that
$I_{0}(t)=aS_{\mathrm{proj}}(t)$, where $I_{0}(t)=10^{-H_{\mathrm{V}}(t)/2.5}$
and its uncertainty is $\Delta I_{0}(t)=\ln(10)I_{0}(t)\Delta V(t)/2.5$ (using
Equation (3) with fixed $G$). Here, $H_{\mathrm{V}}(t)$ is the phase-corrected
absolute magnitude, $H_{\mathrm{V}}$, calculated using Equation (3) at epoch
$t$. We used intensity units rather than magnitudes since uncertainties in
magnitude are not Gaussian, precluding the use of $\chi^{2}$ minimization. The
reduced $\chi^{2}$ statistic is calculated for each case by using

$\chi^{2}_{\mathrm{red}}(\varphi_{1},\varphi_{2})=\frac{1}{N\hskip-2.0pt-\hskip-2.0pt1}\sum_{i=1}^{N}\left(\frac{I_{0}(t_{i})-aS_{\mathrm{proj}}(t_{i};\varphi_{1},\varphi_{2})}{\Delta
I_{0}(t_{i})}\right)^{2},$ (14)

where the denominator $N-1$ is the number of degrees of freedom, $N=9$ is the
number of data points, and $t_{i}$ is the $i$-th observational time (Table 2).
By varying $\varphi_{1}$ and $\varphi_{2}$, we find an $a_{0}$ for which
$a_{0}=\arg\min_{a}\chi^{2}_{\mathrm{red}}(\varphi_{1},\varphi_{2})$. The
minimum $\chi^{2}_{\mathrm{red}}$ is then saved for each model.

The “good” models are selected such that the reduced $\chi^{2}$-statistics is
smaller than $1+\chi^{2}_{\mathrm{red,\,min}}$, where
$\chi^{2}_{\mathrm{red,\,min}}$ is the minimum of such reduced $\chi^{2}$
statistics of all the models. Because we have only one free parameter ($a$),
this gives the models with 1-$\sigma$ confidence interval. The models are then
converted into magnitude system using $-2.5\log_{10}(aI_{0})$. The minimum
$\chi^{2}$ model is the red solid line and other “good” models are black faint
dotted lines in Figure 2. We confirmed that changes of $G$ in the range
$G\in[0.10,\,0.50]$ do not significantly distort the results, except for a
nearly parallel shift caused by the nearly constant change (since all
$\alpha$’s are similar) in all $H_{\mathrm{V}}$ values (Equation 3).

Figure 5: The cross-sectional area seen by an observer at infinite distance
towards the direction of $(\theta_{\mathrm{bf}},\,\phi_{\mathrm{bf}})$,
described both in color and contour. $\theta_{\mathrm{bf}}=0$ is the direction
to the positive $Z$-axis specified by the shape model. Cross-sections are in
units of km2.

##  Appendix B On-Line Material

This work was carried using mostly open-source software, and we here disclose
the codes we used to obtain our results. The original codes are available in a
dedicated GitHub directory999https://github.com/ysBach/KMTNet_Toutatis which
contains five notebooks:

  * •

01_data_reduction: the data reduction process described in Section 2 and the
generation of Table 2.

  * •

02_photometry_comparison: the comparison of our reduced data with Spencer et
al. (1995) and internally developed Moving Object Detection Pipeline (MODP) to
cross-check our photometry results.

  * •

03_radiance_factor: the calculation of $I/F$, $f_{1}$, $f_{2}$, and $A_{5}$
values from the photometry results and projected area calculation (see
Appendix A), and the generation of Figures 2 and 3.

  * •

04_cross_section: the calculation of
$S_{\mathrm{proj}}(\theta_{\mathrm{bf}},\,\phi_{\mathrm{bf}})$ in Figure 5
using ParaView software to check whether the $S_{\mathrm{proj}}(t)$ values in
Appendix A (used IDL POLYSHADE) is reasonable.

  * •

05_polarimetry_plot: the generation of Figure 4 and a short discussion of the
results in Geake & Dollfus (1986).

  * •

06_KMTNet_Seeing: a short description about weather and seeing condition of
our observations in Table 2.

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