11institutetext: Korea Astronomy and Space Science Institute, 776, Daedeokdae-
ro, Yuseong-gu, Daejeon 34055, Korea
11email<EMAIL_ADDRESS>22institutetext: Astronomical Observatory Institute,
Faculty of Physics, A. Mickiewicz University, Słoneczna 36, 60-286 Poznań,
Poland33institutetext: Chungbuk National University, 1 Chungdae-ro, Seowon-gu,
Cheongju, Chungbuk 28644, Korea44institutetext: University of Science and
Technology, 217, Gajeong-ro, Yuseong-gu, Daejeon 34113, Korea55institutetext:
Astronomical Observatory, Jagiellonian University, ul. Orla 171, 30-244
Kraków, Poland66institutetext: Mt. Suhora Observatory, Pedagogical University,
ul. Pochorażych 2, 30-084 Kraków, Poland77institutetext: Las Cumbres
Observatory, 6740 Cortona Drive Suite 102, Goleta, CA 93117,
USA88institutetext: Las Cumbres Observatory, School of Physics and Astronomy,
Cardiff University, QueensBuildings, The Parade, Cardiff CF24 3AA,
UK99institutetext: Asteroid Institute, 20 Sunnyside Ave, Suite 427, Mill
Valley, CA 94941, USA1010institutetext: Department of Astronomy and the DIRAC
Institute, University of Washington, 3910 15thAve NE, Seattle, WA 98195,
USA1111institutetext: Konkoly Observatory, Research Centre for Astronomy and
Earth Sciences, Eőtvős Loránd Research Network (ELKH), H-1121 Budapest,
Konkoly Thege Miklós út 15-17, Hungary1212institutetext: South African
Astronomical Observatory, Cape Town, 7925, South Africa1313institutetext:
Department of Astronomy, University of Cape Town, Rondebosch, 7701, South
Africa1414institutetext: Kepler Institute of Astronomy, University of Zielona
Góra, Lubuska 2, 65-265 Zielona Góra, Poland1515institutetext: Departamento de
Sistema Solar, Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la
Astronomía s/n, 18008 Granada, Spain1616institutetext: National Youth Space
Center, Goheung, Jeollanam-do 59567, Korea1717institutetext: Deutsches Zentrum
für Luft- und Raumfahrt (DLR), Institute of Planetary Research, D-12489
Berlin, Germany1818institutetext: University of Occupational and Environmental
Health, Japan, 1-1 Iseigaoka, Yahata, Kitakyusyu 807-8555,
Japan1919institutetext: Planetary Exploration Research Center, Chiba Institute
of Technology, 2-17-1 Tsudanuma, Narashino, Chiba, 275-0016,
Japan2020institutetext: University of North Carolina at Chapel Hill, Chapel
Hill, North Carolina NC 27599, USA2121institutetext: Department of Physics,
Adiyaman University, 02040 Adiyaman, Turkey2222institutetext: Dark Sky
Observatory, Dept. of Physics and Astronomy, Appalachian State University,
Boone, NC 28608, USA2323institutetext: Akdeniz University, Department of Space
Sciences and Technologies, 07058 Antalya, Turkey2424institutetext: TÜBİTAK
National Observatory, Akdeniz University Campus, 07058 Antalya, Turkey

# Refinement of the convex shape model and tumbling spin state of (99942)
Apophis using the 2020-2021 apparition data

H.-J. Lee 11 M.-J. Kim 11 A. Marciniak 22 D.-H. Kim, 1133 H.-K. Moon 11 Y.-J.
Choi, 1144 S. Zoła 55 J. Chatelain 77 T. A. Lister 77 E. Gomez 88 S.
Greenstreet, 991010 A. Pál 1111 R. Szakáts 1111 N. Erasmus 1212 R. Lees 1313
P. Janse van Rensburg, 12121313
W. Ogłoza 66 M. Dróżdż 66 M. Żejmo 1414 K. Kamiński 22 M. K. Kamińska 22 R.
Duffard 1515 D.-G. Roh 11 H.-S. Yim 11 T. Kim 1616 S. Mottola 1717 F. Yoshida,
18181919 D. E. Reichart 2020 E. Sonbas 2121 D. B. Caton 2222 M. Kaplan 2323 O.
Erece, 23232424 H. Yang 11

(Received ; accepted )

###### Abstract

Context. The close approach of the near-Earth asteroid (99942) Apophis to
Earth in 2029 will provide a unique opportunity to examine how the physical
properties of the asteroid could be changed due to the Earth’s gravitational
perturbation. As a result, the Republic of Korea is planning a rendezvous
mission to Apophis.

Aims. Our aim was to use photometric data from the apparitions in 2020-2021 to
refine the shape model and spin state of Apophis.

Methods. Using thirty-six 1 to 2-m class ground-based telescopes and the
Transiting Exoplanet Survey Satellite, we performed a photometric observation
campaign throughout the 2020-2021 apparition. The convex shape model and spin
state were refined using the light-curve inversion method.

Results. According to our best-fit model, Apophis is rotating in a short axis
mode with rotation and precession periods of 264.178 hours and 27.38547 hours,
respectively. The angular momentum vector orientation of Apophis was found as
(275∘, -85∘) in the ecliptic coordinate system. The ratio of the dynamic
moments of inertia of this asteroid was fitted to
$I_{a}:I_{b}:I_{c}=0.64:0.97:1$, which corresponds to an elongated prolate
ellipsoid. These findings regarding the spin state and shape model could be
used to not only design the space mission scenario but also investigate the
impact of the Earth’s tidal force during close encounters.

###### Key Words.:

Minor planets, asteroids: individual: (99942) Apophis – Techniques:
photometric

Table 1: Details of the observatories and instruments in this campaign.

Telescope | Latitude | Longitude | Instrument
---|---|---|---
–Ground-based telescopes-
Adiyaman Observatory 0.6 m |  37:45:06 N |  38:13:32 E | Andor Tech
AMU Winer, RBT 0.7m |  31:39:56 N | 110:36:06 W | Andor iXon
ATLAS HKO 0.5 m |  20:42:27 N | 156:15:25 W | STA-1600 10.5K CCD
ATLAS MLO 0.5 m |  19:32:10 N | 155:34:34 W | STA-1600 10.5K CCD
BOAO 1.8 m |  36:09:53 N | 128:58:36 E | E2V 4K CCD
CAHA 1.23 m |  37:13:25 N |  2:32:46 W | DLR MKIII camera
CAHA 2.2 m |  37:13:25 N |  2:32:46 W | CAFOS
DOAO 1.0 m |  34:31:35 N | 127:26:48 E | PI SOPHIA-2048B CCD
Kawabe Cosmic Park 1.0 m |  33:53:27 N | 135:13:12 E | FLI PL09000
KMTNet CTIO 1.6 m |  30:10:02 S |  70:48:14 W | 18K mosaic CCD with four E2V 9K
KMTNet SAAO 1.6 m |  32:22:43 S |  20:48:37 E | 18K mosaic CCD with four E2V 9K
KMTNet SSO 1.6 m |  31:16:16 S | 149:03:45 E | 18K mosaic CCD with four E2V 9K
Krakow-CDK500 0.5 m |  50:03:15 N |  19:49:41 E | Apogee USB/Net
LCO CTIO A 1.0 m |  30:10:03 S |  70:48:17 W | Sinistro
LCO CTIO B 1.0 m |  30:10:03 S |  70:48:17 W | Sinistro
LCO McDonald A 1.0 m |  30:41:47 S | 104:00:54 E | Sinistro
LCO McDonald B 1.0 m |  30:41:48 S | 104:00:54 E | Sinistro
LCO SAAO A 1.0 m |  32:23:50 N |  20:48:37 W | Sinistro
LCO SAAO B 1.0 m |  32:23:50 N |  20:48:36 W | Sinistro
LCO SAAO C 1.0 m |  32:23:51 N |  20:48:36 W | Sinistro
LCO SSO A 1.0 m |  31:16:22 N | 149:04:14 W | Sinistro
LCO SSO B 1.0 m |  31:16:23 N | 149:04:15 W | Sinistro
LOAO 1.0 m |  32:26:32 N | 110:47:19 E | E2V 4K CCD
OWL Mitzpeh Ramon 0.5 m |  30:35:51 N |  34:45:48 E | FLI 16803
OWL Oukaimeden 0.5 m |  31:12:21 N |  7:52:00 W | FLI 16803
OWL Tucson 0.5 m |  32:26:31 N | 110:47:21 W | FLI 16803
OWL Yeongcheon 0.5 m |  36:09:50 N | 128:58:33 E | FLI 16803
SAAO Lesedi 1.0 m |  32:22:47 S |  20:48:36 E | SHOC (Andor iXon 888)
Skynet DSO-14, 0.4 m |  36:15:01 N |  81:24:45 W | Apogee USB/Net
Skynet Prompt5, 0.4 m |  30:10:03 S |  70:48:19 W | Apogee USB/Net
Skynet Prompt6, 0.4 m |  30:10:03 S |  70:48:19 W | FLI
Skynet Prompt MO 1 0.4 m |  31:38:18 S | 116:59:19 E | Apogee USB/Net
Skynet RRRT 0.6 m |  37:52:44 N |  78:41:39 W | SBIG STX-16803 3 CCD
SOAO 0.6 m |  36:56:04 N | 128:27:27 E | FLI 4K
Suhora Observatory Zeiss-60 |  49:34:09 N |  20:04:03 E | Apogee AltaU-47
TUG 1.0 m |  36:49:27 N |  30:20:08 E | SI 4K
–Space-based telescope-
TESS 0.1 m |  |  | four MIT Lincoln Lab. CCID-80 devices

111AMU = Adam Mickiewicz University, RBT = Roman Baranowski Telescope, ATLAS =
Asteroid Terrestrial-impact Last Alert System, HKO = Haleakala Observatory,
MLO = Mauna Loa Observatory, BOAO = Bohyunsan Optical Astronomy Observatory,
DOAO = Deokheung Optical Astronomy Observatory, CAHA = Calar Alto Observatory,
KMTNet = Korea Microlensing Telescope (Kim et al., 2016), CTIO = Cerro Tololo
Inter-American Observatory, SAAO = South African Astronomical Observatory, SSO
= Siding Spring Observatory, LCO = Las Cumbres Observatory, LOAO = Lemonsan
Optical Astronomy Observatory, CDK = Corrected Dall-Kirkham, OWL = Optical
Wide-field patroL Network, DSO = Dark Sky Observatory, MO = Meckering
Observatory, RRRT = Rapid Response Robotic Telescope, SOAO = Sobaeksan Optical
Astronomy Observatory, TUG = TÜBÍTAK National observatory, TESS = Transiting
Exoplanet Survey Satellite, CAFOS = Calar Alto Faint Object Spectrograph

##  1 Introduction

The Aten-type near-Earth asteroid (99942) Apophis (2004 MN4) (hereafter
Apophis) is an Sq-type asteroid with an estimated size of 340 m (Binzel et
al., 2009; Brozović et al., 2018; Reddy et al., 2018). This asteroid was
discovered on June 19, 2004, by R.A. Tucker, D.J. Tholen, and F. Bernardi at
Kitt Peak, Arizona. From early prediction, it was estimated that this asteroid
could impact Earth with a maximum probability of 2.7% (JPL Sentry on December
27, 2004; Chesley 2006). As the accuracy of the orbital prediction improved in
follow-up observations, the impact probability of Apophis was reduced. In
particular, the prediction derived from high-precision radar observations at
the Arecibo Observatory in 2005 and 2006 indicated that this asteroid will
pass by Earth at a geocentric distance of 38,326 km (approximately 6 Earth
radii), which is within the geosynchronous orbit in April 2029 (Giorgini et
al., 2008). Recently, the radar observations at the Goldstone Solar System
Radar (GSSR) and Green Bank Telescope (GBT) in March 2021 have precisely
estimated Apophis’ orbit around the Sun, ruling out any Earth impact threat
for the next hundred years or more (Greicius, 2021).

Although Apophis’ impact threat has disappeared, this asteroid remains an
object of interest because of its close approach in 2029. The Earth encounter
of Apophis in 2029 is expected to trigger varying degrees of alterations in
the dynamics, spin-states, and surface arrangements of Apophis due to the
Earth’s gravitational perturbation (Yu et al., 2014; Souchay et al., 2014,
2018; DeMartini et al., 2019; Hirabayashi et al., 2021; Valvano et al., 2022).
Thus, the study of this asteroid will provide an excellent opportunity to
examine the evolutionary process of its physical properties caused by
planetary perturbation. For this reason, Apophis became a unique observation
target and is the primary mission target of the Rendezvous Mission to Apophis,
which is currently under pre-Phase A study in the Republic of Korea and
scheduled for launch in 2027 (Moon et al., 2020).

The shape model and spin state are the most fundamental parameters for
predicting the evolutionary process due to Earth’s tidal effect. In addition,
these properties provide important information for planning space mission
scenarios. The spin state and convex shape model of Apophis were reconstructed
using photometric data obtained from the 2012-2013 apparition (Pravec et al.,
2014). They found it has non-principal axis (NPA) rotation in a short-axis
mode (SAM) with rotation and precession periods of 263 and 27.38 hours,
respectively, and the orientation of angular momentum vector of $\lambda_{L}$
= 250∘ and $\beta_{L}$ = -75∘. In addition, the convex shape model of Pravec
et al. (2014) can be approximated by a prolate ellipsoid with a ratio of the
greatest and intermediate principal moments of inertia ($I_{b}/I_{c}$) of
0.965, and the smallest principal moment of inertia ($I_{a}/I_{c}$) of 0.61.

For the last time before the 2029 Earth encounter, Apophis approached Earth on
March 2021 at 0.11 AU and its apparent brightness increased to reach magnitude
16. Thus, the observation window for Apophis from the end of 2020 to the
beginning of 2021 was the last opportunity to investigate its spin properties
and refine the convex shape model. Therefore, we planned a photometric
observation campaign for Apophis during this apparition. The details of our
observation campaign are described in Section 2. In Section 3, a period
analysis and reconstruction of the spin state and shape model of Apophis are
reported. Finally, the summary and conclusions of this letter are given in
Section 4.

Figure 1: $P_{2}$ search diagram of Apophis. The sum of square residuals were
calculated for the 4-th order two-period Fourier series with $P_{1}$ = 30.56
hours fitted to our dense light curve obtained from 2021-02-07.0 to
2021-03-16.2 in flux units. Figure 2: The light curve of Apophis taken from
2021-02-07.0 to 2021-03-16.2 reduced to the unit geo- and heliocentric
distance and to a consistent solar phase angle. The blue open circles indicate
the photometric data folded with $P_{1}$. The red curves denote the best-fit
4th order two-period Fourier series with the periods $P_{1}$ = 30.56 hours and
$P_{2}$ = 27.38 hours. The black squares indicate the residuals of the
photometric data from the 4th order two-period Fourier series (see the right
ordinates). Figure 3: Convex shape model of Apophis.

##  2 Photometric observation campaign in the 2020-2021 apparition

As mentioned above, the 2020-2021 apparition of Apophis was the last
opportunity to reveal the physical properties of this asteroid before its 2029
close approach. Therefore, we organized an extensive and long-term photometric
observation campaign for Apophis during this apparition. Our observation
campaign was conducted by using not only the ground-based telescope but also
the space telescope that is Transiting Exoplanet Survey Satellite (TESS;
Ricker et al., 2015) spacecraft. The details of telescopes and instruments
used in our campaign are provided in Table 1. The geometries and observational
circumstances are listed in Table LABEL:table:A1. This campaign’s data also
contributed to the global Apophis Planetary Defense Campaign (Reddy et al.,
2022).

Ground-based observations were carried out in 11 countries, including the
Republic of Korea, the US, Chile, South Africa, Australia, Poland, Spain,
Turkey, and Japan using 36 telescopes. Through our observation campaign, we
observed 214 dense-in-time light curves and two sparse-in-time light curves.
The dense-in-time light curves were obtained using a Johnson-Cousins $V$ or
$R$ filter except for the data from the Kawabe Cosmic Park, which used the
SDSS $r^{\prime}$ filter. As the observations were made with different
telescopes and instruments, the data reduction processes may be slightly
different. All data reductions were conducted following the standard manner.
The bias- and flat-field images corrected, and the instrument magnitudes for
each frame were measured using aperture photometry. All photometric data were
calibrated with ATLAS Refcat2 (The ATLAS All-Sky Stellar Reference Catalog;
Tonry et al., 2018a). The magnitudes of ATLAS Refcat2 were converted to
Johnson–Cousins $V$ and $R$ magnitudes using empirical transformation
equations (Tonry et al., 2012). On the other hand, the sparse-in-time light
curves were obtained from ATLAS (Tonry et al., 2018a, b). The ATLAS light
curve was observed between November 2020 and April 2021 using orange ($o$,
560–820 nm) and cyan ($c$, 420-650 nm) filters.

We also gathered the long-term continuous photometric data observed from the
TESS spacecraft using a wide $I$ passband (see Figure 1 in Ricker et al.
(2015)). TESS photometric data have been obtained in a similar manner as it
was done in (Pál et al., 2020) for the image series of Sector 35 as acquired
between 2021-02-19 and 2021-03-07. The individual photometric data points were
derived using a convolution-based differential image analysis by employing the
tools of the FITSH package Pál (2012).

##  3 Results

###  3.1 Periodic analysis

Before the shape model and spin state of Apophis were reconstructed, we
performed periodic analysis of the light curve obtained from our observation
campaign. To minimize the possible systematic effects caused by the changes in
the observing geometry, this analysis was only conducted using the dense-in-
time light curves observed with phase angles from 20∘ to 40∘, which
corresponds to the period from 2021-02-07.0 to 2021-03-16.2. Because our
observations were made with different filters, we corrected them to match the
R-filter using the color indices $V-R$ = 0.38, $R-T$ = 0.07, and
$r^{\prime}-R$ = 0.33. The data were converted to flux units, and the helio-
and geocentric distances were corrected to the unit distance and the solar
phase angle to a consistent value using the H-G phase relation assuming G =
0.24. As Apophis exhibited tumbling motion, we attempted to detect double
periods from the light curve of this asteroid.

First, the Lomb-Scargle method (Lomb, 1976; Scargle, 1982) was adopted to
search for periodicities in the light curve. The strongest signal in the Lomb-
Scargle periodogram was found at a period of 15.28 hours. The Lomb-Scargle
method performs periodic analysis based on a single peak light curve. However,
most light curves of asteroids have double peaks because their shapes are
elongated. Therefore, we determined the primary period ($P_{1}$) on the light
curve of Apophis as 30.56 hours that is double the strongest signal in the
Lomb-Scargle periodogram.

The secondary period ($P_{2}$) was determined using the two-period Fourier
series method (Pravec et al., 2005, 2014). The two-period Fourier series
employed in this analysis are presented as follows:

$\displaystyle F(t)=C_{0}+\sum\limits_{j=1}^{m}\Bigg{[}C_{j0}\cos{\frac{2\pi
j}{P_{1}}}t+S_{j0}\sin{\frac{2\pi j}{P_{1}}}t\Bigg{]}$
$\displaystyle\qquad\;+\sum\limits_{k=1}^{m}\sum\limits_{j=-m}^{m}\Bigg{[}C_{jk}\cos{(\frac{2\pi
j}{P_{1}}+\frac{2\pi k}{P_{2}})}t+S_{jk}\sin{(\frac{2\pi j}{P_{1}}+\frac{2\pi
k}{P_{2}})}t\Bigg{]}$

The result for the $P_{2}$ search is shown in Figure 1, where the abscissa is
$P_{2}$ and the ordinate is the sum of the squared residuals for the fitted
4-th order two-period Fourier series with $P_{1}$ = 30.56 hours. From our
$P_{2}$ search, we found two minimums at 27.38 and 263 hours. As the long
period of 263 hours was a combination of the short period of 27.38 hours with
$P_{1}$, that is, $263^{-1}\approx 27.38^{-1}-30.56^{-1}$, it seems to be
derived from the short period. Thus, we determined $P_{2}$ as 27.38 hours.
Figure 2 shows a composite light curve of Apophis obtained from 2021-02-07.0
to 2021-03-16.2 with the fitted 4-th order two-period Fourier series for
periods 30.56 and 27.38 hours.

###  3.2 Reconstruction of the convex shape model and spin state from light
curves

Table 2: Parameters of the Apophis model.

| This Work | Pravec et al. (2014)
---|---|---
$\lambda_{L}$ [deg] | $278^{+9}_{-8}$ | $250\pm 27$
$\beta_{L}$ [deg] | $-86^{+5}_{-4}$ | $-75\pm 14$
$P_{\psi}$ [hour] | $264.18\pm 0.03$ | $263\pm 6$
$P_{\phi}$ [hour] | $27.3855\pm 0.0003$ | $27.38\pm 0.07$
$\psi_{0}$a [deg] | $3^{+5}_{-1}$ | $14^{+44}_{-11}$
$\phi_{0}$a [deg] | $183^{+7}_{-4}$ | $152^{+173}_{-64}$
$I_{a}/I_{c}$ | $0.64^{+0.02}_{-0.09}$ | $0.61^{+0.11}_{-0.08}$
$I_{b}/I_{c}$ | $0.962^{+0.023}_{-0.002}$ | $0.965^{+0.009}_{-0.015}$
$a_{\mathrm{dyn}}/c_{\mathrm{dyn}}$ | $1.48\pm 0.19$ | 1.51 $\pm$ 0.18
$b_{\mathrm{dyn}}/c_{\mathrm{dyn}}$ | $1.06\pm 0.04$ | 1.06 $\pm$ 0.02
$a_{\mathrm{shape}}/c_{\mathrm{shape}}$ | $1.56\pm 0.04$ | 1.64 $\pm$ 0.09
$b_{\mathrm{shape}}/c_{\mathrm{shape}}$ | $1.12\pm 0.03$ | $1.14^{+0.04}_{-0.08}$
$E/E_{0}$ | $1.018\pm 0.010$ | $1.024\pm 0.013$
222$\lambda_{L}$ and $\beta_{L}$ are the ecliptic coordinates of the constant
angular momentum vector $\mathbf{L}$; $\psi_{0}$ and $\phi_{0}$ are the
standard Euler angles at epoch JD0; $I_{a}$, $I_{b}$ and $I_{c}$ are the
dynamical moments of inertia corresponding to the longest-, intermediate-, and
shortest-axes; $a_{\mathrm{dyn}}/c_{\mathrm{dyn}}$ and
$c_{\mathrm{dyn}}/c_{\mathrm{dyn}}$ are the axial ratios of a dynamically
equivalent ellipsoid; $a_{\mathrm{shape}}/c_{\mathrm{shape}}$ and
$c_{\mathrm{shape}}/c_{\mathrm{shape}}$ are the axial ratios of a convex shape
model; $E/E_{0}$ is the ratio of the rotational kinetic energy to the lowest
energy for the given angular momentum.
a The epoch JD0 is 2456284.676388 (2012-12-23.176388).

Given that the radar observation of Apophis suggested that it has a bifurcated
shape (Brozović et al., 2018), the non-convex shape model may be a more
suitable description of its actual shape. Nonetheless, this non-convex model
obtained using photometric data should be applied only after very careful
consideration. It is generally not possible to uniquely reconstruct a non-
convex model using only photometric data (Viikinkoski et al., 2017; Harris &
Warner, 2020). Further, concavities can be revealed only when reconstruction
is performed using data observed at sufficiently high phase angles (Ďurech &
Kaasalainen, 2003). Unfortunately, we did not obtain sufficient data at high
phase angles to create a non-convex model of Apophis. Therefore, our analysis
was conducted based on the convex shape model.

The convex shape model and spin state of Apophis were reconstructed using the
light curve inversion method for the NPA rotator (Kaasalainen & Torppa, 2001;
Kaasalainen et al., 2001; Kaasalainen, 2001) combined with Hapke’s light-
scattering model (Hapke, 1993). In this work, we used not only the light
curves obtained through our campaign observation but also all available
observation data collected from the literature. The historical light curves
are listed in Table LABEL:table:A2.

The first step in revealing the spin state of a tumbling asteroid is to
determine the physical periods, that is, the rotation ($P_{\psi}$) and
precession period ($P_{\phi}$). Because the periods on the light curve were
derived from the physical periods, $P_{1}^{-1}$ and $P_{2}^{-1}$ usually
appear at $P_{\phi}^{-1}$ and $P_{\phi}^{-1}\pm P_{\psi}^{-1}$, where the plus
sign is for the long-axis mode (LAM) and the minus sign for SAM (Kaasalainen,
2001). Therefore, we found two possible physical period combinations using the
period obtained in the previous section: $P_{1}^{-1}=P_{\phi}^{-1}$ and
$P_{2}^{-1}=P_{\phi}^{-1}+P_{\psi}^{-1}$ (LAM);
$P_{1}^{-1}=P_{\phi}^{-1}-P_{\psi}^{-1}$ and $P_{2}^{-1}=P_{\phi}^{-1}$ (SAM).
The optimization for each $P_{\phi}$ was performed in the same way as in Lee
et al. (2020, 2021). In these produces, the Hapke’s model parameters were set
to a typical S-type asteroid’s: $\varpi=0.23$, $g=-0.27$, $h=0.08$,
$B_{0}=1.6$, and $\bar{\theta}=20^{\circ}$ (Li et al., 2015). Because we did
not obtain the data observed at low solar phase angles, the parameters for the
opposition surge, $h$ and $B_{0}$, and the roughness $\bar{\theta}$ were
fixed. Only the $\varpi$ and $g$ parameters were optimized. It was found that
the inertia tensor of the convex shape model for the LAM solution was not
consistent with the kinematic $I_{1}$ and $I_{2}$ parameters. Therefore, we
decided to use SAM as the final solution.

The best-fit model for Apophis is listed in Table 2, together with Pravec et
al. (2014) solution for comparison. Further, the uncertainties of our model
parameters corresponded to a 3 $\sigma$ confidence interval. The confidence
interval was estimated from the increase in the $\chi^{2}$ value when the
solved-for physical parameters were varied. The threshold corresponding to the
3 $\sigma$ confidence interval was set as
$\chi^{2}_{min}\times(1+3\sqrt{2/\nu})$333These was a typo in Vokrouhlický et
al. (2017), so we used $2/\nu$ instead of $2\nu$., where the $\chi^{2}_{min}$
represents the $\chi^{2}$ value for the best-fit solution and $\nu$ represents
the number of degrees of freedom (Vokrouhlický et al., 2017). The convex shape
model of Apophis is shown in Figure 3. The synthetic light curve of our model
and the observed light curve are presented in Appendix B.

##  4 Summary and Conclusions

In this letter, we present the convex shape model and spin state of Apophis
reconstructed using historical and newly obtained light curves. The new
photometric data were observed from an extensive and long-term photometric
observation campaign for Apophis during its 2020-2021 apparition using 36
facilities, including the ground-based and space telescope. We obtained 211
dense-in-time light curves and two spare-in-time light curves. In the period
analysis conducted with our dense light curve, double periods were detected as
30.56 and 27.38 hours, respectively.

The best-fit solution indicates that Apophis is in a SAM state with rotation
and precession periods of 264.178 $\pm$ 0.01 and 27.38547 $\pm$ 0.00002 hours,
respectively. The ecliptic coordinates of the angular momentum vector
orientation of this asteroid are (275$\pm$3∘, -85$\pm$1∘). In addition, the
ratio of the dynamical moments of inertia was estimated as $I_{a}/I_{c}$ =
0.64 and $I_{b}/I_{c}$ = 0.96. Our model was similar to that of Pravec et al.
(2014). Nonetheless, the uncertainties of the model parameters were improved
because they were reconstructed based on the dataset obtained from the two
apparitions. This model will be useful not only for investigating changes in
Apophis’ physical properties due to the tidal effect during its encounter in
2029 but also for planning a space mission to this asteroid.

###### Acknowledgements.

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.
This paper was partially based on observations obtained at the Bohyunsan
Optical Astronomy Observatory(BOAO), the Sobaeksan Optical Astronomy
Observatory(SOAO), and the Lemmonsan Optical Astronomy Observatory(LOAO),
which is operated by the Korea Astronomy and Space Science Institute (KASI).
This work has made use of data from the Asteroid Terrestrial-impact Last Alert
System (ATLAS) project. The Asteroid Terrestrial-impact Last Alert System
(ATLAS) project is primarily funded to search for near earth asteroids through
NASA grants NN12AR55G, 80NSSC18K0284, and 80NSSC18K1575; byproducts of the NEO
search include images and catalogs from the survey area. The ATLAS science
products have been made possible through the contributions of the University
of Hawaii Institute for Astronomy, the Queen’s University Belfast, the Space
Telescope Science Institute, the South African Astronomical Observatory, and
The Millennium Institute of Astrophysics (MAS), Chile. A.P. and R.S. were
supported by the K-138962 grant of the National Research, Development and
Innovation Office. R.D. acknowledge financial support from the State Agency
for Research of the Spanish MCIU through the “Center of Excellence Severo
Ochoa” award for the Instituto de Astrofísica de Andalucía (SEV-2017-0709).
Based on observations collected at Centro Astronómico Hispano en Andalucía
(CAHA) at Calar Alto, operated jointly by Instituto de Astrofísica de
Andalucía (CSIC) and Junta de Andalucía.

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##  Appendix A Additional tables

Table 3: The geometries and observational circumstances. Date UT | R. A. | Dec. | $r_{h}$ | $\Delta$ | $\alpha$ | Observatory | Filter
---|---|---|---|---|---|---|---
| (h m) | (∘ ’) | (AU) | (AU) | (∘) |  |
-–Dense photometry–- |
2021-01-16.5 | 11 44 | -17 26 | 1.076 | 0.203 | 58.2 | LOAO | $R$
2021-01-17.5 | 11 43 | -17 37 | 1.077 | 0.200 | 57.5 | LOAO | $R$
2021-01-18.5 | 11 43 | -17 48 | 1.078 | 0.198 | 56.7 | LOAO | $R$
2021-01-29.1 | 11 34 | -19 16 | 1.091 | 0.170 | 48.1 | SAAO Lesedi | $VR$
2021-02-05.0 | 11 21 | -19 33 | 1.096 | 0.153 | 41.4 | SAAO Lesedi | $V$
2021-02-05.4 | 11 20 | -19 33 | 1.096 | 0.152 | 41.0 | Skynet Prompt5 | $R$
2021-02-06.0 | 11 19 | -19 31 | 1.096 | 0.151 | 40.3 | SAAO Lesedi | $V$
2021-02-06.3 | 11 18 | -19 31 | 1.096 | 0.150 | 40.0 | OWL USA | $R$
2021-02-07.0 | 11 16 | -19 29 | 1.097 | 0.148 | 39.3 | CAHA 2.2 m | $R$
2021-02-07.1 | 11 16 | -19 28 | 1.097 | 0.148 | 39.2 | LCO SAAO A | $R$
2021-02-07.1 | 11 16 | -19 28 | 1.097 | 0.148 | 39.2 | LCO SAAO B | $R$
2021-02-07.1 | 11 16 | -19 28 | 1.097 | 0.148 | 39.2 | SAAO Lesedi | $V$
2021-02-07.2 | 11 16 | -19 28 | 1.097 | 0.148 | 39.1 | LCO CTIO A | $R$
2021-02-07.3 | 11 15 | -19 28 | 1.097 | 0.148 | 38.9 | OWL USA | $R$
2021-02-07.3 | 11 15 | -19 28 | 1.097 | 0.148 | 38.9 | Skynet Prompt5 | $R$
2021-02-07.8 | 11 14 | -19 26 | 1.097 | 0.147 | 38.4 | LCO SAAO C | $R$
2021-02-08.0 | 11 14 | -19 25 | 1.097 | 0.146 | 38.2 | LCO SAAO A | $R$
2021-02-08.0 | 11 14 | -19 25 | 1.097 | 0.146 | 38.2 | LCO SAAO B | $R$
2021-02-08.1 | 11 13 | -19 24 | 1.097 | 0.146 | 38.1 | SAAO Lesedi | $V$
2021-02-08.3 | 11 13 | -19 23 | 1.097 | 0.145 | 37.8 | LCO CTIO A | $R$
2021-02-08.3 | 11 13 | -19 23 | 1.097 | 0.145 | 37.8 | LCO McDonald A | $R$
2021-02-08.3 | 11 13 | -19 23 | 1.097 | 0.145 | 37.8 | OWL USA | $R$
2021-02-08.3 | 11 13 | -19 23 | 1.097 | 0.145 | 37.8 | Skynet Prompt5 | $R$
2021-02-08.6 | 11 12 | -19 22 | 1.097 | 0.145 | 37.5 | LCO SSO B | $R$
2021-02-08.9 | 11 11 | -19 20 | 1.098 | 0.144 | 37.2 | LCO SAAO A | $R$
2021-02-09.1 | 11 10 | -19 19 | 1.098 | 0.144 | 37.0 | LCO SAAO B | $R$
2021-02-09.1 | 11 10 | -19 19 | 1.098 | 0.144 | 37.0 | SAAO Lesedi | $V$
2021-02-09.3 | 11 10 | -19 17 | 1.098 | 0.143 | 36.7 | LCO McDonald B | $R$
2021-02-09.3 | 11 10 | -19 17 | 1.098 | 0.143 | 36.7 | Skynet Prompt5 | $R$
2021-02-09.4 | 11 10 | -19 17 | 1.098 | 0.143 | 36.6 | OWL USA | $R$
2021-02-09.6 | 11 09 | -19 15 | 1.098 | 0.142 | 36.4 | LCO SSO B | $R$
2021-02-09.9 | 11 08 | -19 13 | 1.098 | 0.142 | 36.1 | LCO SAAO C | $R$
2021-02-10.0 | 11 08 | -19 12 | 1.098 | 0.142 | 35.9 | LCO SAAO A | $R$
2021-02-10.1 | 11 07 | -19 12 | 1.098 | 0.141 | 35.8 | LCO SAAO B | $R$
2021-02-10.1 | 11 07 | -19 12 | 1.098 | 0.141 | 35.8 | OWL MAR | $R$
2021-02-10.1 | 11 07 | -19 12 | 1.098 | 0.141 | 35.8 | SAAO Lesedi | $V$
2021-02-10.2 | 11 07 | -19 11 | 1.098 | 0.141 | 35.7 | Skynet Prompt5 | $R$
2021-02-10.3 | 11 07 | -19 10 | 1.098 | 0.141 | 35.6 | LCO CTIO B | $R$
2021-02-10.4 | 11 07 | -19 09 | 1.098 | 0.141 | 35.5 | LCO McDonald B | $R$
2021-02-10.5 | 11 06 | -19 08 | 1.098 | 0.140 | 35.4 | LCO SSO B | $R$
2021-02-11.0 | 11 05 | -19 04 | 1.098 | 0.139 | 34.8 | Adiyaman Observatory | $R$
2021-02-11.0 | 11 05 | -19 04 | 1.098 | 0.139 | 34.8 | LCO SAAO C | $R$
2021-02-11.0 | 11 05 | -19 04 | 1.098 | 0.139 | 34.8 | OWL ISR | $R$
2021-02-11.1 | 11 04 | -19 03 | 1.098 | 0.139 | 34.7 | OWL MAR | $R$
2021-02-11.2 | 11 04 | -19 02 | 1.098 | 0.139 | 34.6 | LCO CTIO A | $R$
2021-02-11.3 | 11 04 | -19 01 | 1.098 | 0.139 | 34.5 | LCO CTIO B | $R$
2021-02-11.3 | 11 04 | -19 01 | 1.098 | 0.139 | 34.5 | Skynet Prompt5 | $R$
2021-02-11.4 | 11 03 | -19 00 | 1.098 | 0.139 | 34.3 | Winer Observatory RBT | $R$
2021-02-11.5 | 11 03 | -18 59 | 1.098 | 0.138 | 34.2 | LCO SSO A | $R$
2021-02-11.5 | 11 03 | -18 59 | 1.098 | 0.138 | 34.2 | OWL USA | $R$
2021-02-12.0 | 11 01 | -18 54 | 1.099 | 0.137 | 33.7 | Adiyaman Observatory | $R$
2021-02-12.2 | 11 01 | -18 52 | 1.099 | 0.137 | 33.4 | OWL MAR | $R$
2021-02-12.3 | 11 00 | -18 51 | 1.099 | 0.137 | 33.3 | Winer Observatory RBT | $R$
2021-02-12.4 | 11 00 | -18 50 | 1.099 | 0.136 | 33.2 | OWL USA | $R$
2021-02-12.7 | 10 59 | -18 46 | 1.099 | 0.136 | 32.9 | Skynet Prompt MO1 | $R$
2021-02-13.2 | 10 57 | -18 40 | 1.099 | 0.135 | 32.3 | Skynet Prompt5 | $R$
2021-02-14.0 | 10 54 | -18 29 | 1.099 | 0.133 | 31.4 | Adiyaman Observatory | $R$
2021-02-15.1 | 10 50 | -18 13 | 1.099 | 0.131 | 30.1 | CAHA 2.2 m | $R$
2021-02-16.1 | 10 47 | -17 55 | 1.099 | 0.129 | 29.0 | OWL MAR | $R$
2021-02-16.4 | 10 45 | -17 50 | 1.099 | 0.129 | 28.6 | Skynet Prompt5 | $R$
2021-02-16.6 | 10 45 | -17 46 | 1.099 | 0.128 | 28.4 | Skynet Prompt MO1 | $R$
2021-02-16.9 | 10 43 | -17 40 | 1.099 | 0.128 | 28.1 | OWL MAR | $R$
2021-02-17.4 | 10 41 | -17 30 | 1.099 | 0.127 | 27.5 | Winer Observatory RBT | $R$
2021-02-17.7 | 10 40 | -17 24 | 1.099 | 0.127 | 27.2 | Skynet Prompt MO1 | $R$
2021-02-18.4 | 10 37 | -17 09 | 1.099 | 0.125 | 26.5 | LOAO | $R$
2021-02-18.4 | 10 37 | -17 09 | 1.099 | 0.125 | 26.5 | Winer Observatory RBT | $R$
2021-02-18.7 | 10 36 | -17 02 | 1.099 | 0.125 | 26.1 | SOAO | $R$
2021-02-18.8 | 10 36 | -16 60 | 1.099 | 0.125 | 26.0 | Skynet Prompt MO1 | $R$
2021-02-19.0 | 10 35 | -16 55 | 1.099 | 0.125 | 25.8 | Suhora Observatory | $R$
2021-02-19.3 | 10 34 | -16 48 | 1.099 | 0.124 | 25.5 | LOAO | $R$
2021-02-19.3 | 10 34 | -16 48 | 1.099 | 0.124 | 25.5 | Winer Observatory RBT | $R$
2021-02-19.6 | 10 32 | -16 40 | 1.099 | 0.124 | 25.2 | BOAO | $R$
2021-02-19.6 | 10 32 | -16 40 | 1.099 | 0.124 | 25.2 | OWL KOR | $R$
2021-02-19.6 | 10 32 | -16 40 | 1.099 | 0.124 | 25.2 | SOAO | $R$
2021-02-19.7 | 10 32 | -16 38 | 1.099 | 0.123 | 25.2 | Skynet Prompt MO1 | $R$
2021-02-20.3 | 10 29 | -16 23 | 1.099 | 0.123 | 24.6 | LOAO | $R$
2021-02-20.5 | 10 28 | -16 17 | 1.099 | 0.122 | 24.4 | Skynet Prompt MO1 | $R$
2021-02-20.6 | 10 28 | -16 15 | 1.099 | 0.122 | 24.3 | BOAO | $R$
2021-02-21.0 | 10 26 | -16 04 | 1.099 | 0.122 | 24.0 | Suhora Observatory | $R$
2021-02-21.4 | 10 25 | -15 52 | 1.099 | 0.121 | 23.6 | Winer Observatory RBT | $R$
2021-02-21.6 | 10 24 | -15 47 | 1.099 | 0.121 | 23.5 | BOAO | $R$
2021-02-21.8 | 10 23 | -15 41 | 1.099 | 0.121 | 23.3 | OWL ISR | $R$
2021-02-22.5 | 10 20 | -15 20 | 1.099 | 0.120 | 22.8 | Skynet Prompt MO1 | $R$
2021-02-22.7 | 10 19 | -15 14 | 1.098 | 0.119 | 22.6 | BOAO | $R$
2021-02-22.7 | 10 19 | -15 14 | 1.098 | 0.119 | 22.6 | Skynet Prompt MO1 | $R$
2021-02-23.1 | 10 17 | -15 01 | 1.098 | 0.119 | 22.4 | OWL USA | $R$
2021-02-24.2 | 10 12 | -14 25 | 1.098 | 0.118 | 21.8 | OWL USA | $R$
2021-02-25.0 | 10 08 | -13 58 | 1.098 | 0.117 | 21.4 | Suhora Observatory | $R$
2021-02-26.2 | 10 03 | -13 14 | 1.097 | 0.116 | 21.1 | OWL USA | $R$
2021-02-27.2 | 09 58 | -12 36 | 1.097 | 0.115 | 21.1 | OWL USA | $R$
2021-02-28.0 | 09 54 | -12 04 | 1.096 | 0.115 | 21.2 | CAHA 2.2 m | $R$
2021-02-28.0 | 09 54 | -12 04 | 1.096 | 0.115 | 21.2 | Skynet Prompt5 | $R$
2021-03-01.2 | 09 49 | -11 14 | 1.096 | 0.114 | 21.5 | OWL USA | $R$
2021-03-01.9 | 09 45 | -10 45 | 1.095 | 0.114 | 21.9 | TUG | $R$
2021-03-03.0 | 09 40 | -09 57 | 1.095 | 0.113 | 22.6 | OWL MAR | $R$
2021-03-03.0 | 09 40 | -09 57 | 1.095 | 0.113 | 22.6 | Suhora Observatory | $R$
2021-03-03.0 | 09 40 | -09 57 | 1.095 | 0.113 | 22.6 | KMTNet CTIO | $VR$
2021-03-03.0 | 09 40 | -09 57 | 1.095 | 0.113 | 22.6 | Skynet Prompt5 | $R$
2021-03-03.2 | 09 39 | -09 48 | 1.095 | 0.113 | 22.7 | OWL USA | $R$
2021-03-03.2 | 09 39 | -09 48 | 1.095 | 0.113 | 22.7 | Winer Observatory RBT | $R$
2021-03-03.4 | 09 39 | -09 39 | 1.095 | 0.113 | 22.9 | KMTNet SSO | $VR$
2021-03-05.1 | 09 31 | -08 21 | 1.093 | 0.113 | 24.4 | LCO CTIO A | $R$
2021-03-05.2 | 09 30 | -08 16 | 1.093 | 0.113 | 24.5 | LCO McDonald A | $R$
2021-03-05.4 | 09 30 | -08 07 | 1.093 | 0.113 | 24.7 | LCO SSO B | $R$
2021-03-05.6 | 09 29 | -07 58 | 1.093 | 0.113 | 24.9 | LCO SSO A | $R$
2021-03-05.8 | 09 28 | -07 48 | 1.093 | 0.113 | 25.1 | LCO SAAO A | $R$
2021-03-05.8 | 09 28 | -07 48 | 1.093 | 0.113 | 25.1 | LCO SAAO C | $R$
2021-03-06.4 | 09 25 | -07 20 | 1.092 | 0.113 | 25.8 | LCO SSO B | $R$
2021-03-06.5 | 09 25 | -07 15 | 1.092 | 0.113 | 25.9 | LCO SSO A | $R$
2021-03-06.8 | 09 23 | -07 01 | 1.092 | 0.113 | 26.3 | LCO SAAO C | $R$
2021-03-06.8 | 09 23 | -07 01 | 1.092 | 0.113 | 26.3 | OWL ISR | $R$
2021-03-06.9 | 09 23 | -06 56 | 1.092 | 0.113 | 26.4 | LCO SAAO A | $R$
2021-03-06.9 | 09 23 | -06 56 | 1.092 | 0.113 | 26.4 | OAUJ CDK500 | $R$
2021-03-07.0 | 09 22 | -06 51 | 1.092 | 0.113 | 26.5 | LCO CTIO B | $R$
2021-03-07.1 | 09 22 | -06 47 | 1.092 | 0.113 | 26.6 | OWL USA | $R$
2021-03-07.2 | 09 22 | -06 42 | 1.092 | 0.113 | 26.8 | LCO McDonald A | $R$
2021-03-07.3 | 09 21 | -06 37 | 1.092 | 0.113 | 26.9 | LCO McDonald B | $R$
2021-03-07.4 | 09 21 | -06 32 | 1.092 | 0.113 | 27.0 | KMTNet SSO | $VR$
2021-03-07.5 | 09 20 | -06 27 | 1.091 | 0.113 | 27.1 | LCO SSO A | $R$
2021-03-07.6 | 09 20 | -06 23 | 1.091 | 0.113 | 27.3 | LCO SSO B | $R$
2021-03-07.8 | 09 19 | -06 13 | 1.091 | 0.113 | 27.5 | KMTNet SAAO | $VR$
2021-03-07.8 | 09 19 | -06 13 | 1.091 | 0.113 | 27.5 | LCO SAAO A | $R$
2021-03-07.9 | 09 19 | -06 08 | 1.091 | 0.113 | 27.6 | LCO SAAO C | $R$
2021-03-08.0 | 09 18 | -06 03 | 1.091 | 0.113 | 27.8 | LCO CTIO B | $R$
2021-03-08.1 | 09 18 | -05 58 | 1.091 | 0.113 | 27.9 | LCO McDonald A | $R$
2021-03-08.2 | 09 17 | -05 54 | 1.091 | 0.113 | 28.0 | LCO McDonald B | $R$
2021-03-08.8 | 09 15 | -05 25 | 1.090 | 0.113 | 28.8 | OWL ISR | $R$
2021-03-08.9 | 09 14 | -05 20 | 1.090 | 0.113 | 29.0 | LCO SAAO C | $R$
2021-03-09.0 | 09 14 | -05 15 | 1.090 | 0.113 | 29.1 | LCO CTIO B | $R$
2021-03-09.0 | 09 14 | -05 15 | 1.090 | 0.113 | 29.1 | LCO SAAO B | $R$
2021-03-09.5 | 09 12 | -04 51 | 1.090 | 0.113 | 29.8 | LCO SSO A | $R$
2021-03-09.6 | 09 12 | -04 46 | 1.090 | 0.113 | 29.9 | LCO SSO B | $R$
2021-03-09.6 | 09 12 | -04 46 | 1.090 | 0.113 | 29.9 | SOAO | $R$
2021-03-09.8 | 09 11 | -04 36 | 1.089 | 0.113 | 30.2 | KMTNet SAAO | $VR$
2021-03-09.8 | 09 11 | -04 36 | 1.089 | 0.113 | 30.2 | LCO SAAO B | $R$
2021-03-10.0 | 09 10 | -04 26 | 1.089 | 0.113 | 30.5 | Suhora Observatory | $R$
2021-03-10.2 | 09 09 | -04 16 | 1.089 | 0.114 | 30.8 | LCO CTIO A | $R$
2021-03-10.2 | 09 09 | -04 16 | 1.089 | 0.114 | 30.8 | Skynet DSO-14 | $R$
2021-03-10.3 | 09 09 | -04 12 | 1.089 | 0.114 | 30.9 | OWL USA | $R$
2021-03-10.3 | 09 09 | -04 12 | 1.089 | 0.114 | 30.9 | Skynet RRRT | $R$
2021-03-10.3 | 09 09 | -04 12 | 1.089 | 0.114 | 30.9 | Winer Observatory RBT | $R$
2021-03-10.5 | 09 08 | -04 02 | 1.089 | 0.114 | 31.2 | BOAO | $R$
2021-03-10.5 | 09 08 | -04 02 | 1.089 | 0.114 | 31.2 | DOAO | $R$
2021-03-10.6 | 09 08 | -03 57 | 1.089 | 0.114 | 31.3 | SOAO | $R$
2021-03-10.6 | 09 08 | -03 57 | 1.089 | 0.114 | 31.3 | Kawabe Observatory | $r^{\prime}$
2021-03-10.9 | 09 06 | -03 42 | 1.088 | 0.114 | 31.8 | CAHA 1.23 m | $R$
2021-03-10.9 | 09 06 | -03 42 | 1.088 | 0.114 | 31.8 | OAUJ CDK500 | $R$
2021-03-11.0 | 09 06 | -03 37 | 1.088 | 0.114 | 31.9 | Suhora Observatory | $R$
2021-03-11.1 | 09 06 | -03 33 | 1.088 | 0.114 | 32.1 | LCO CTIO B | $R$
2021-03-11.2 | 09 05 | -03 28 | 1.088 | 0.114 | 32.2 | Skynet RRRT | $R$
2021-03-11.3 | 09 05 | -03 23 | 1.088 | 0.114 | 32.4 | OWL USA | $R$
2021-03-11.3 | 09 05 | -03 23 | 1.088 | 0.114 | 32.4 | Winer Observatory RBT | $R$
2021-03-11.8 | 09 03 | -02 59 | 1.087 | 0.114 | 33.1 | LCO SAAO A | $R$
2021-03-11.9 | 09 02 | -02 54 | 1.087 | 0.114 | 33.2 | LCO SAAO B | $R$
2021-03-11.9 | 09 02 | -02 54 | 1.087 | 0.114 | 33.2 | LCO SAAO C | $R$
2021-03-12.0 | 09 02 | -02 49 | 1.087 | 0.114 | 33.4 | LCO CTIO B | $R$
2021-03-12.2 | 09 01 | -02 39 | 1.087 | 0.115 | 33.7 | LCO CTIO A | $R$
2021-03-12.3 | 09 01 | -02 34 | 1.087 | 0.115 | 33.8 | Winer Observatory RBT | $R$
2021-03-12.6 | 08 60 | -02 20 | 1.086 | 0.115 | 34.3 | LCO SSO A | $R$
2021-03-12.7 | 08 60 | -02 15 | 1.086 | 0.115 | 34.4 | LCO SAAO B | $R$
2021-03-12.9 | 08 59 | -02 05 | 1.086 | 0.115 | 34.7 | LCO SAAO A | $R$
2021-03-12.9 | 08 59 | -02 05 | 1.086 | 0.115 | 34.7 | OAUJ CDK500 | $R$
2021-03-12.9 | 08 59 | -02 05 | 1.086 | 0.115 | 34.7 | Suhora Observatory | $R$
2021-03-13.2 | 08 58 | -01 51 | 1.086 | 0.115 | 35.2 | LCO CTIO B | $R$
2021-03-13.6 | 08 56 | -01 36 | 1.086 | 0.115 | 35.6 | DOAO | $R$
2021-03-13.8 | 08 56 | -01 22 | 1.085 | 0.116 | 36.1 | TUG | $R$
2021-03-13.9 | 08 55 | -01 17 | 1.085 | 0.116 | 36.2 | LCO SAAO A | $R$
2021-03-14.0 | 08 55 | -01 12 | 1.085 | 0.116 | 36.4 | Skynet Prompt5 | $R$
2021-03-14.0 | 08 55 | -01 12 | 1.085 | 0.116 | 36.4 | Skynet RRRT | $R$
2021-03-14.1 | 08 54 | -01 07 | 1.085 | 0.116 | 36.5 | LCO CTIO B | $R$
2021-03-14.3 | 08 54 | +00 58 | 1.085 | 0.116 | 36.8 | LCO McDonald A | $R$
2021-03-14.5 | 08 53 | +00 48 | 1.084 | 0.116 | 37.1 | BOAO | $R$
2021-03-15.0 | 08 51 | +00 25 | 1.084 | 0.116 | 37.9 | OAUJ CDK500 | $R$
2021-03-15.0 | 08 51 | +00 25 | 1.084 | 0.116 | 37.9 | Skynet Prompt5 | $R$
2021-03-15.1 | 08 51 | +00 20 | 1.084 | 0.117 | 38.0 | Skynet Prompt6 | $R$
2021-03-15.1 | 08 51 | +00 20 | 1.084 | 0.117 | 38.0 | Skynet RRRT | $R$
2021-03-15.2 | 08 51 | +00 15 | 1.084 | 0.117 | 38.2 | LOAO | $R$
2021-03-15.9 | 08 48 | +00 18 | 1.083 | 0.117 | 39.3 | OAUJ CDK500 | $R$
2021-03-16.0 | 08 48 | +00 22 | 1.083 | 0.117 | 39.4 | CAHA 2.2 m | $R$
2021-03-16.2 | 08 47 | +00 32 | 1.082 | 0.118 | 39.7 | LOAO | $R$
2021-03-16.6 | 08 46 | +00 50 | 1.082 | 0.118 | 40.3 | BOAO | $R$
2021-03-16.6 | 08 46 | +00 50 | 1.082 | 0.118 | 40.3 | DOAO | $R$
2021-03-16.9 | 08 45 | +01 04 | 1.081 | 0.118 | 40.8 | Adiyaman Observatory | $R$
2021-03-17.0 | 08 45 | +01 09 | 1.081 | 0.118 | 40.9 | CAHA 2.2 m | $R$
2021-03-17.0 | 08 45 | +01 09 | 1.081 | 0.118 | 40.9 | Skynet Prompt5 | $R$
2021-03-17.1 | 08 45 | +01 13 | 1.081 | 0.118 | 41.1 | Skynet Prompt6 | $R$
2021-03-17.6 | 08 43 | +01 32 | 1.081 | 0.119 | 41.7 | DOAO | $R$
2021-03-18.0 | 08 42 | +01 55 | 1.080 | 0.119 | 42.4 | CAHA 2.2 m | $R$
2021-03-18.2 | 08 41 | +02 04 | 1.080 | 0.119 | 42.7 | OWL USA | $R$
2021-03-18.4 | 08 41 | +02 13 | 1.079 | 0.120 | 43.0 | KMTNet SSO | $VR$
2021-03-18.8 | 08 40 | +02 31 | 1.079 | 0.120 | 43.6 | KMTNet SAAO | $VR$
2021-03-18.8 | 08 40 | +02 31 | 1.079 | 0.120 | 43.6 | OWL ISR | $R$
2021-03-19.2 | 08 38 | +02 48 | 1.078 | 0.120 | 44.2 | OWL USA | $R$
2021-03-19.9 | 08 37 | +03 19 | 1.077 | 0.121 | 45.2 | OWL ISR | $R$
2021-03-20.2 | 08 36 | +03 32 | 1.077 | 0.122 | 45.7 | OWL USA | $R$
2021-03-20.2 | 08 36 | +03 32 | 1.077 | 0.122 | 45.7 | Skynet RRRT | $R$
2021-03-21.2 | 08 33 | +04 15 | 1.076 | 0.123 | 47.1 | OWL USA | $R$
2021-03-22.3 | 08 30 | +05 01 | 1.074 | 0.124 | 48.7 | OWL USA | $R$
2021-03-22.5 | 08 30 | +05 10 | 1.074 | 0.124 | 49.0 | DOAO | $R$
2021-03-22.8 | 08 29 | +05 22 | 1.073 | 0.125 | 49.4 | KMTNet SAAO | $VR$
2021-03-24.8 | 08 25 | +06 42 | 1.070 | 0.127 | 52.2 | KMTNet SAAO | $VR$
2021-03-30.0 | 08 16 | +09 53 | 1.061 | 0.134 | 59.0 | KMTNet CTIO | $VR$
2021-03-30.7 | 08 15 | +10 16 | 1.060 | 0.135 | 59.9 | KMTNet SAAO | $VR$
2021-04-01.8 | 08 12 | +11 25 | 1.055 | 0.138 | 62.5 | KMTNet SAAO | $VR$
2021-04-03.1 | 08 11 | +12 06 | 1.053 | 0.139 | 64.1 | KMTNet CTIO | $VR$
2021-04-04.9 | 08 09 | +13 00 | 1.049 | 0.142 | 66.2 | CAHA 1.23 m | $R$
2021-04-05.4 | 08 08 | +13 15 | 1.048 | 0.142 | 66.7 | KMTNet SSO | $VR$
2021-04-05.9 | 08 08 | +13 30 | 1.047 | 0.143 | 67.3 | CAHA 1.23 m | $R$
2021-04-08.4 | 08 06 | +14 39 | 1.042 | 0.146 | 70.1 | KMTNet SSO | $VR$
2021-04-08.8 | 08 06 | +14 50 | 1.041 | 0.147 | 70.6 | KMTNet SAAO | $VR$
2021-04-14.8 | 08 02 | +17 22 | 1.026 | 0.154 | 77.1 | KMTNet SAAO | $VR$
2021-04-16.8 | 08 02 | +18 09 | 1.021 | 0.156 | 79.2 | KMTNet SAAO | $VR$
2021-04-18.4 | 08 01 | +18 45 | 1.017 | 0.158 | 80.8 | KMTNet SSO | $VR$
2021-04-18.8 | 08 01 | +18 54 | 1.016 | 0.158 | 81.2 | KMTNet SAAO | $VR$
2021-04-20.4 | 08 01 | +19 29 | 1.012 | 0.160 | 82.9 | KMTNet SSO | $VR$
–Sparse Photometry- |
2020-11-12 2021-04-09 |  |  |  | ATLAS | $c$
2020-12-01 2021-04-03 |  |  |  | ATLAS | $o$
–Space-based Photometry- |
2021-02-19 2021-03-07 |  |  |  | TESS | wide $I$
Table 3: continued. 444R. A. : Right ascension, Dec. : Declination, $\Delta$ :
geocentric distance, $r_{h}$ : heliocentric distance, $\alpha$ : phase angle.

Table 4: List of the historical light curves. Date UT | R. A. | Dec. | $r_{h}$ | $\Delta$ | $\alpha$ | References
---|---|---|---|---|---|---
| (h m) | (∘ ’) | (AU) | (AU) | (∘) |
2012-12-23.3 | 10 42 | -27 22 | 1.000 | 0.102 | 77.6 | Pravec et al. (2014)
2012-12-25.3 | 10 33 | -27 23 | 1.006 | 0.101 | 74.3 | Pravec et al. (2014)
2012-12-26.3 | 10 28 | -27 23 | 1.009 | 0.101 | 72.7 | Pravec et al. (2014)
2012-12-27.2 | 10 23 | -27 21 | 1.011 | 0.101 | 71.2 | Pravec et al. (2014)
2012-12-28.2 | 10 18 | -27 19 | 1.014 | 0.100 | 69.5 | Pravec et al. (2014)
2012-12-29.2 | 10 13 | -27 14 | 1.017 | 0.100 | 67.8 | Pravec et al. (2014)
2012-12-30.2 | 10 08 | -27 10 | 1.019 | 0.099 | 66.2 | Pravec et al. (2014)
2012-12-31.2 | 10 03 | -27 03 | 1.022 | 0.099 | 64.5 | Pravec et al. (2014)
2013-01-03.2 | 09 47 | -26 34 | 1.029 | 0.098 | 59.5 | Pravec et al. (2014)
2013-01-04.2 | 09 41 | -26 20 | 1.032 | 0.097 | 57.9 | Pravec et al. (2014)
2013-01-05.2 | 09 36 | -26 06 | 1.034 | 0.097 | 56.2 | Pravec et al. (2014)
2013-01-06.1 | 09 30 | -25 51 | 1.036 | 0.097 | 54.7 | Pravec et al. (2014)
2013-01-06.2 | 09 30 | -25 49 | 1.036 | 0.097 | 54.6 | Pravec et al. (2014)
2013-01-07.2 | 09 24 | -25 30 | 1.039 | 0.097 | 52.9 | Pravec et al. (2014)
2013-01-07.6 | 09 22 | -25 22 | 1.040 | 0.097 | 52.3 | Pravec et al. (2014)
2013-01-08.1 | 09 19 | -25 11 | 1.041 | 0.097 | 51.5 | Pravec et al. (2014)
2013-01-08.1 | 09 19 | -25 11 | 1.041 | 0.097 | 51.5 | Pravec et al. (2014)
2013-01-08.2 | 09 18 | -25 10 | 1.041 | 0.097 | 51.3 | Pravec et al. (2014)
2013-01-09.1 | 09 13 | -24 49 | 1.043 | 0.097 | 49.9 | Pravec et al. (2014)
2013-01-09.1 | 09 13 | -24 49 | 1.043 | 0.097 | 49.9 | Pravec et al. (2014)
2013-01-09.2 | 09 12 | -24 46 | 1.043 | 0.097 | 49.7 | Pravec et al. (2014)
2013-01-09.3 | 09 12 | -24 44 | 1.043 | 0.097 | 49.5 | Pravec et al. (2014)
2013-01-10.1 | 09 07 | -24 24 | 1.045 | 0.097 | 48.3 | Pravec et al. (2014)
2013-01-10.2 | 09 06 | -24 22 | 1.045 | 0.097 | 48.1 | Pravec et al. (2014)
2013-01-10.3 | 09 06 | -24 19 | 1.046 | 0.097 | 48.0 | Pravec et al. (2014)
2013-01-11.1 | 09 01 | -23 57 | 1.047 | 0.097 | 46.7 | Pravec et al. (2014)
2013-01-11.3 | 09 00 | -23 52 | 1.048 | 0.097 | 46.4 | Pravec et al. (2014)
2013-01-11.5 | 08 59 | -23 46 | 1.048 | 0.097 | 46.1 | Pravec et al. (2014)
2013-01-12.1 | 08 55 | -23 28 | 1.049 | 0.097 | 45.2 | Pravec et al. (2014)
2013-01-12.2 | 08 55 | -23 25 | 1.050 | 0.097 | 45.1 | Pravec et al. (2014)
2013-01-12.2 | 08 55 | -23 25 | 1.050 | 0.097 | 45.1 | Pravec et al. (2014)
2013-01-13.1 | 08 49 | -22 57 | 1.051 | 0.097 | 43.7 | Pravec et al. (2014)
2013-01-13.2 | 08 49 | -22 54 | 1.052 | 0.097 | 43.6 | Pravec et al. (2014)
2013-01-14.1 | 08 44 | -22 24 | 1.053 | 0.097 | 42.3 | Pravec et al. (2014)
2013-01-14.1 | 08 44 | -22 24 | 1.053 | 0.097 | 42.3 | Pravec et al. (2014)
2013-01-14.2 | 08 43 | -22 20 | 1.054 | 0.097 | 42.2 | Pravec et al. (2014)
2013-01-15.1 | 08 38 | -21 49 | 1.055 | 0.098 | 40.9 | Pravec et al. (2014)
2013-01-15.2 | 08 37 | -21 45 | 1.056 | 0.098 | 40.8 | Pravec et al. (2014)
2013-01-15.5 | 08 35 | -21 34 | 1.056 | 0.098 | 40.4 | Pravec et al. (2014)
2013-01-16.1 | 08 32 | -21 12 | 1.057 | 0.098 | 39.6 | Pravec et al. (2014)
2013-01-16.1 | 08 32 | -21 12 | 1.057 | 0.098 | 39.6 | Pravec et al. (2014)
2013-01-16.3 | 08 31 | -21 04 | 1.058 | 0.098 | 39.4 | Pravec et al. (2014)
2013-01-16.3 | 08 31 | -21 04 | 1.058 | 0.098 | 39.4 | Pravec et al. (2014)
2013-01-19.1 | 08 15 | -19 11 | 1.063 | 0.100 | 36.2 | Pravec et al. (2014)
2013-01-19.2 | 08 15 | -19 07 | 1.063 | 0.100 | 36.1 | Pravec et al. (2014)
2013-01-20.1 | 08 10 | -18 28 | 1.065 | 0.101 | 35.2 | Pravec et al. (2014)
2013-01-20.2 | 08 10 | -18 24 | 1.065 | 0.101 | 35.1 | Pravec et al. (2014)
2013-01-22.2 | 07 59 | -16 54 | 1.068 | 0.103 | 33.5 | Pravec et al. (2014)
2013-01-22.3 | 07 59 | -16 50 | 1.068 | 0.103 | 33.5 | Pravec et al. (2014)
2013-01-23.1 | 07 55 | -16 13 | 1.070 | 0.104 | 33.0 | Pravec et al. (2014)
2013-01-23.2 | 07 55 | -16 08 | 1.070 | 0.104 | 32.9 | Pravec et al. (2014)
2013-01-24.1 | 07 50 | -15 26 | 1.071 | 0.105 | 32.4 | Pravec et al. (2014)
2013-01-24.2 | 07 50 | -15 22 | 1.071 | 0.105 | 32.4 | Pravec et al. (2014)
2013-01-24.3 | 07 49 | -15 17 | 1.072 | 0.105 | 32.4 | Pravec et al. (2014)
2013-02-04.1 | 07 10 | -06 46 | 1.086 | 0.123 | 33.8 | Pravec et al. (2014)
2013-02-05.1 | 07 07 | -06 01 | 1.087 | 0.125 | 34.4 | Pravec et al. (2014)
2013-02-05.3 | 07 07 | -05 53 | 1.087 | 0.126 | 34.5 | Pravec et al. (2014)
2013-02-06.1 | 07 05 | -05 17 | 1.088 | 0.128 | 35.0 | Pravec et al. (2014)
2013-02-06.2 | 07 04 | -05 13 | 1.088 | 0.128 | 35.1 | Pravec et al. (2014)
2013-02-06.9 | 07 03 | -04 43 | 1.089 | 0.129 | 35.5 | Pravec et al. (2014)
2013-02-07.1 | 07 02 | -04 34 | 1.089 | 0.130 | 35.7 | Pravec et al. (2014)
2013-02-07.2 | 07 02 | -04 29 | 1.089 | 0.130 | 35.7 | Pravec et al. (2014)
2013-02-08.1 | 07 00 | -03 51 | 1.090 | 0.132 | 36.4 | Pravec et al. (2014)
2013-02-08.2 | 07 00 | -03 47 | 1.090 | 0.132 | 36.4 | Pravec et al. (2014)
2013-02-12.1 | 06 53 | -01 09 | 1.093 | 0.142 | 39.3 | Pravec et al. (2014)
2013-02-13.1 | 06 52 | -00 31 | 1.094 | 0.145 | 40.1 | Pravec et al. (2014)
2013-02-13.2 | 06 52 | -00 27 | 1.094 | 0.145 | 40.2 | Pravec et al. (2014)
2013-02-14.2 | 06 50 | +00 10 | 1.094 | 0.148 | 40.9 | Pravec et al. (2014)
2013-02-14.2 | 06 50 | +00 10 | 1.094 | 0.148 | 40.9 | Pravec et al. (2014)
2013-02-15.1 | 06 49 | +00 43 | 1.095 | 0.150 | 41.6 | Pravec et al. (2014)
2013-02-15.1 | 06 49 | +00 43 | 1.095 | 0.150 | 41.6 | Pravec et al. (2014)
2013-02-16.0 | 06 48 | +01 15 | 1.095 | 0.152 | 42.3 | Pravec et al. (2014)
2013-02-16.2 | 06 48 | +01 22 | 1.096 | 0.153 | 42.5 | Pravec et al. (2014)
2013-02-16.2 | 06 48 | +01 22 | 1.096 | 0.153 | 42.5 | Pravec et al. (2014)
2013-02-17.0 | 06 48 | +01 49 | 1.096 | 0.155 | 43.0 | Pravec et al. (2014)
2013-02-17.2 | 06 47 | +01 56 | 1.096 | 0.156 | 43.2 | Pravec et al. (2014)
2013-02-17.2 | 06 47 | +01 56 | 1.096 | 0.156 | 43.2 | Pravec et al. (2014)
2013-02-18.0 | 06 47 | +02 23 | 1.096 | 0.158 | 43.8 | Pravec et al. (2014)
2013-02-18.0 | 06 47 | +02 23 | 1.096 | 0.158 | 43.8 | Pravec et al. (2014)
2013-02-18.2 | 06 47 | +02 30 | 1.096 | 0.159 | 44.0 | Pravec et al. (2014)
2013-02-19.9 | 06 46 | +03 25 | 1.097 | 0.163 | 45.2 | Pravec et al. (2014)
2013-03-09.1 | 06 49 | +10 38 | 1.096 | 0.215 | 56.3 | Pravec et al. (2014)
2013-03-10.2 | 06 50 | +11 00 | 1.095 | 0.218 | 56.9 | Pravec et al. (2014)
2013-03-11.0 | 06 51 | +11 15 | 1.095 | 0.221 | 57.3 | Pravec et al. (2014)
2013-03-12.0 | 06 52 | +11 33 | 1.094 | 0.224 | 57.9 | Pravec et al. (2014)
2013-03-13.0 | 06 53 | +11 51 | 1.094 | 0.227 | 58.4 | Pravec et al. (2014)
2013-04-09.0 | 07 33 | +17 20 | 1.060 | 0.297 | 70.7 | Pravec et al. (2014)
2013-04-12.0 | 07 38 | +17 43 | 1.054 | 0.303 | 71.9 | Pravec et al. (2014)
2013-04-14.1 | 07 42 | +17 57 | 1.050 | 0.307 | 72.8 | Pravec et al. (2014)
2013-04-15.0 | 07 44 | +18 03 | 1.048 | 0.308 | 73.2 | Pravec et al. (2014)
2021-03-18.3 | 08 41 | +02 08 | 1.080 | 0.120 | 42.9 | Warner & Stephens (2021)
2021-03-19.2 | 08 38 | +02 48 | 1.078 | 0.120 | 44.2 | Warner & Stephens (2021)
2021-03-20.3 | 08 35 | +03 37 | 1.077 | 0.122 | 45.8 | Warner & Stephens (2021)
2021-03-21.2 | 08 33 | +04 15 | 1.076 | 0.123 | 47.1 | Warner & Stephens (2021)
2021-03-22.3 | 08 30 | +05 01 | 1.074 | 0.124 | 48.7 | Warner & Stephens (2021)
2021-03-23.3 | 08 28 | +05 42 | 1.072 | 0.125 | 50.1 | Warner & Stephens (2021)
2021-03-24.2 | 08 26 | +06 19 | 1.071 | 0.126 | 51.4 | Warner & Stephens (2021)
2021-03-27.2 | 08 20 | +08 13 | 1.066 | 0.130 | 55.4 | Warner & Stephens (2021)
2021-03-28.2 | 08 19 | +08 50 | 1.064 | 0.131 | 56.7 | Warner & Stephens (2021)
2021-03-29.2 | 08 17 | +09 25 | 1.062 | 0.133 | 58.0 | Warner & Stephens (2021)
2021-03-30.2 | 08 16 | +09 59 | 1.060 | 0.134 | 59.3 | Warner & Stephens (2021)
2021-03-31.2 | 08 14 | +10 33 | 1.059 | 0.135 | 60.5 | Warner & Stephens (2021)
Table 4: continued. 555R. A. : Right ascension, Dec. : Declination, $\Delta$ :
geocentric distance, $r_{h}$ : heliocentric distance, $\alpha$ : phase angle.

##  Appendix B Light curves

Figure 4: The photometric data in 2012-2013 (black open circle) with the
synthetic light curve from best-fit model (red line). Figure 5: The
photometric data in 2020-2021 (black open circle) with the synthetic light
curve from best-fit model (red line).


# Precision Doppler Shift Measurements with a Frequency Comb Calibrated Laser
Heterodyne Radiometer

Ryan K. Cole  Time and Frequency Division, National Institute of Standards and
Technology, Boulder CO, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>Connor Fredrick  Time and Frequency Division, National Institute of Standards
and Technology, Boulder CO, USA  Department of Physics, University of Colorado
Boulder, Boulder, CO, USA  Newton H. Nguyen  Division of Geological and
Planetary Sciences, California Institute of Technology, Pasadena, CA, USA
Scott A. Diddams  Time and Frequency Division, National Institute of Standards
and Technology, Boulder CO, USA  Department of Physics, University of Colorado
Boulder, Boulder, CO, USA  Department of Electrical, Computer, and Energy
Engineering, University of Colorado Boulder, Boulder, CO, USA

###### Abstract

We report precision atmospheric spectroscopy of CO2 using a laser heterodyne
radiometer (LHR) calibrated with an optical frequency comb. Using the comb-
calibrated LHR, we record spectra of atmospheric CO2 near 1572.33 nm with a
spectral resolution of 200 MHz using sunlight as a light source. The measured
CO2 spectra exhibit frequency shifts by approximately 11 MHz over the course
of the five-hour measurement, and we show that these shifts are caused by
Doppler effects due to wind along the spectrometer line of sight. The measured
frequency shifts are in excellent agreement with an atmospheric model, and we
show that our measurements track the wind-induced Doppler shifts with a
relative frequency precision of 100 kHz ($\mathrm{15\ cm\cdot s^{-1}}$),
equivalent to a fractional precision of a few parts in 1010. These results
demonstrate that frequency-comb-calibrated LHR enables precision velocimetry
that can be of use in applications ranging from climate science to astronomy.

††journal: opticajournal

Laser heterodyne radiometry (LHR) is a well known approach for spectroscopy of
thermal light [1]. In LHR, light from a continuous wave laser (the local
oscillator, LO) is interfered with light from a thermal source, and the
resulting heterodyne signal gives a measure of the power of the thermal light
within a narrow frequency range around the LO laser. By tuning the LO laser
frequency, a high-resolution optical spectrum (e.g. $R=\nu/\delta\nu\sim
10^{6}$) can be recorded within the scan range of the laser without the use of
moving components or diffractive optics.

Numerous past studies have demonstrated LHR with sunlight to record spectra of
atmospheric trace gases [2, 3, 4, 5, 6, 7, 8, e.g.] or to study absorption
transitions in the sun itself [9, 10]. Recently, Fredrick et al. [10]
introduced LHR with a frequency comb calibration, bringing the absolute
stability and traceability of the frequency comb to high-precision
spectroscopy of solar absorption lines and transitions in a laboratory gas
cell. Here, we extend this frequency-comb-calibrated LHR approach to
atmospheric spectroscopy of greenhouse gases, and we show that the high
spectral resolution and frequency precision of comb-calibrated LHR enables
tracking of wind-induced Doppler shifts in the measured spectra with
$\mathrm{cm\cdot s^{-1}}$ precision.

While LHR is a well established technique for measuring mixing ratios of
greenhouse gases and other atmospheric trace gases [2, 3, 4, 5, 6, 7, 8,
e.g.], several studies have also demonstrated that LHR is capable of
atmospheric wind measurements through the Doppler shifts imparted by wind
along the spectrometer line of sight [11, 12, 13]. Atmospheric wind
measurements are relevant in applications ranging from meteorology [14] to
climate and greenhouse gas monitoring [15]. For example, wind drives the
transport of atmospheric greenhouse gases, and when combined with coincident
mixing ratio data, wind speed measurements provide an important constraint in
our understanding of the spatiotemporal gradients of greenhouse gases and
other atmospheric trace gases [15]. To this end, expanding the remote sensing
capabilities of LHR to include atmospheric wind measurements could provide
valuable climate and meteorological data to complement measurements based on
more established techniques (e.g. Doppler radar, lidar or microwave
radiometry). More broadly, extending the capabilities of LHR for Doppler
velocimetry could expand the utility of LHR in applications beyond climate and
meteorology such as precision Doppler spectroscopy of astronomical sources
[16] or passive tracking of thermal objects.

Figure 1: Panel (a): Schematic of the frequency-comb-calibrated LHR approach
(see text for description). Panel (b): Five-hour-averaged spectrum of the CO2
R16 transition near 1572.33 nm. The inset shows the calibration tick marks
(spaced by $f_{r}/2=125\ \textrm{MHz}$) measured by interfering the LO laser
with the frequency comb.

Spectroscopic wind measurements pose a demanding challenge for the
spectrometers used to make the measurement. For example, resolving a Doppler
shift due to 1 $\mathrm{m\cdot s^{-1}}$ line of sight motion requires a
spectrometer with fractional frequency precision ($\delta f/f$) better than
$10^{-9}$. Recent LHR-based wind measurements have addressed this challenge
with a frequency calibration based on an etalon [5] or Mach-Zehnder
interferometer [12] in combination with a reference gas cell that is used to
determine the line center of the target transition in the rest frame. With
this approach, these studies have reported vertically-resolved measurements of
absolute wind speeds with precision at the meter-per-second level [11, 12].
Here, we address the challenge of frequency stability by calibrating our LHR
system with a laser frequency comb to enable spectroscopy of sunlight with the
stability and frequency accuracy of a frequency comb. Using this approach, we
track wind-induced Doppler shifts in measured spectra with precision better
than 100 kHz ($\sim$15 $\mathrm{cm\cdot s^{-1}}$).

Figure 1 shows a schematic of our frequency-comb-calibrated LHR approach. This
apparatus has been described in detail in Ref. [10], and here we list only the
salient details. We couple solar light into single-mode fiber using a solar-
tracking telescope. The telescope consists of a commercial solar tracker (EKO
STR-22G) and piezo-actuated steering mirror that directs solar light onto a
fiber collimator. The steering mirror provides secondary pointing corrections
to account for small deviations in the solar tracking. The steering mirror
pointing is locked to the bright center of the solar disk with feedback based
on the fiber-coupled solar power measured after splitting the solar light in a
1310/1550 nm WDM. A refractive beam shaper placed between the steering mirror
and the fiber collimator uniformly integrates light from the solar disk by
transforming the Gaussian fiber mode to a flat-top profile in the far field
[10].

Fiber-coupled solar light is combined with light from a DFB diode laser (the
LO) that is temperature-tuned over the target absorption transition. The solar
and LO light are combined in a polarization-maintaining 50:50 fiber coupler
and interfered on a balanced photodetector (Thorlabs PDB465A). The radio
frequency (RF) output of the photodetector is sent to an RF power detection
circuit (described below), while the DC monitor output is used to feed back to
a variable optical attenuator (VOA) that stabilizes the LO laser power and
mitigates signal distortion due to variations in laser power during each scan.

The RF power detection circuit is the same as described in Ref. [10]. Briefly,
the heterodyne output of the photodetector is amplified and passed through a
low pass filter (LPF) that sets the spectral resolution of the measurement as
twice the filter cutoff frequency. The filtered signal is split in a power
splitter and passed to both inputs of a double-balanced mixer. The mixer
output is terminated into 50 ohms, and the resulting DC voltage is
proportional to the heterodyne signal power. The DC signal is passed through a
preamplifier and additional low pass filter before being digitized on an
oscilloscope.

In a second channel, the LO light is simultaneously interfered with light from
a stabilized, $f_{r}=250\ \mathrm{MHz}$ Er:fiber frequency comb. The
heterodyne signal between the LO and comb is recorded on a balanced detector
and mixed with a synthesized 62.5 MHz tone that doubles the density of the
frequency calibration points [10, 17]. The RF power detection circuit is the
same as described above, but uses a lower filter cutoff (2 MHz) that limits
the heterodyne signal to a narrow range around each comb mode. The output of
this process is a series of calibration "ticks" that occur whenever the
scanning LO laser coincides with a comb mode.

The output of the comb-calibrated LHR system described above is a DC signal
proportional to the spectrum of the solar light and a simultaneously recorded
series of frequency calibration ticks. We determine the frequency axis of the
measured spectra by fitting each calibration tick with a Gaussian profile to
determine its centroid. Using the resulting calibration points as well as the
known frequency spacing between each point ($f_{R}/2$), we construct a time-
to-frequency transfer function that transforms the temporal axis of the
measurement to the comb-referenced frequency grid. The frequency comb used for
this comb calibration is referenced to a NIST-calibrated hydrogen maser and
provides a SI-traceable frequency calibration grid with relative uncertainty
of a few parts in $10^{13}$ or better. Accounting for the relative uncertainty
in the maser comb reference and the time-to-frequency calibration process, we
estimate the relative frequency uncertainty of the comb calibration to be
$\sim$70 kHz for a single measurement (10 s), averaging to $\sim$5 kHz at one
hour. At that level, line center determination is limited by noise in the
measured spectra.

Figure 2: Panel (a): Comparison between the measured frequency shifts and the
shifts predicted using the atmospheric model. The lower panel shows the
differences between measured and modeled shifts. Measurement times are
specified relative 10:10 a.m. local time (UTC-6). Panel (b): Allan deviation
of the difference between the measured and modeled frequency shifts.

Using the approach outlined above, we recorded spectra for atmospheric CO2 in
Boulder, CO, USA on October 12, 2022. The measurement targeted the R16
transition of the $30012\leftarrow 00001$ CO2 band near 1572.33 nm, which has
been the subject of past remote sensing missions [18] and advanced
spectroscopic characterization [19]. Figure 1(b) shows the measured CO2
spectrum after averaging for nearly five hours. The spectrum was recorded
using a low pass filter bandwidth of 100 MHz, which results in a measurement
spectral resolution of 200 MHz. The effective averaging time (2 ms) of the
final low pass filter in the RF detection chain (see Figure 1) yields $\sim$30
independent samples per 200 MHz resolution element. Each spectrum was recorded
in a 10 s scan spanning a $\sim$30 GHz optical window. The signal-to-noise
ratio (SNR) for each 10 s scan is $\sim$50\. Owing to the high stability of
the comb calibration, long-term averaging of the measured spectra allows the
SNR to grow with $\sqrt{\tau}$, exceeding 2000 after averaging for the full
measurement period.

We assess the relative frequency precision of the measured spectra by
comparing each measurement to a template generated from the five-hour-averaged
spectrum. The observed shift in each spectrum is taken as the frequency shift
that maximizes the cross correlation between the spectrum and the template. In
this sense, this approach determines frequency shifts relative to the spectrum
averaged over the full measurement period. For the CO2 spectra measured on
October 12, the frequency shifts indicate a progressive blue shift by $\sim$11
MHz over the course of the five-hour measurement. Figure 2(a) shows the
measured frequency shifts along with a comparison to our model for the
expected frequency shifts due to wind-induced Doppler effects along the LHR
line of sight.

To model the effect of wind on the measured spectra, atmospheric temperature,
pressure, and three-dimensional wind fields are obtained from the European
Center for Medium-Range Weather Forecast ERA5 reanalysis data [20] for
Boulder, CO, USA. The ERA5 data has a temporal resolution of one hour, and we
linearly interpolate the data to estimate the atmospheric conditions for each
measured LHR spectrum. We split the atmosphere into 50 altitude bins, and
simulate the CO2 R16 transition in each layer using the HITRAN2020 database
with temperature-dependent line shape parameters for the speed-dependent
Nelkin-Ghatak profile (SDNGP) [21]. We assume a uniform CO2 mixing ratio of
400 ppm, which after integrating over the 50 atmospheric layers produces a
simulated line shape that is in qualitative agreement with measured spectra.

Our model accounts for wind-induced Doppler effects by applying a frequency
shift to the simulated spectrum in each atmospheric layer. The wind speed
along the LHR line of sight (and therefore the Doppler shift) is determined as

$\mathcal{W}^{LOS}_{i}=\mathbfcal{W}_{i}\cdot\mathbf{\hat{k}}$ (1)

where $\mathcal{W}^{LOS}_{i}$ is the line-of-sight wind speed in layer $i$,
$\mathbf{\hat{k}}$ is the normalized LHR pointing vector, and
$\mathbfcal{W}_{i}$ is the wind velocity vector in terms of eastward
($\mathbf{\hat{u}}$), northward ($\mathbf{\hat{v}}$), and downward
($\mathbf{\hat{w}}$) components. The pointing vector is specified in the
($u,v,w$) coordinate system as

$\mathbf{k}=\left(sin\theta_{z}sin\alpha\right)\mathbf{\hat{u}}+\left(sin\theta_{z}cos\alpha\right)\mathbf{\hat{v}}+\left(cos\theta_{z}\right)\mathbf{\hat{w}}$
(2)

where $\theta_{z}$ is the solar zenith angle and $\alpha$ is the azimuth
angle. We determine the solar position angles for each measured LHR spectrum
using a Python wrapper for NREL’s Solar Position Algorithm [22, 23]. Figure 3
shows the line of sight wind speeds calculated using this method for the data
on October 12.

After simulating spectra at times corresponding to each measured LHR spectrum,
we determine the wind-induced Doppler shifts using the same cross correlation
approach described above. In this case, the wind-induced shifts are determined
relative to a template generated by averaging the simulated spectra over the
five-hour measurement period. Figure 2(a) shows the wind-induced shifts
calculated using our model, which are in excellent agreement with the measured
shifts over the full duration of the measurement. Figure 2(b) shows the Allan
deviation of the difference between the measured and calculated shifts. For a
single spectrum (10 s), we track the line center with a precision of $\sim$2
MHz (3 $\mathrm{m\cdot s^{-1}}$) and the precision improves to approximately
$\sim$100 kHz (15 $\mathrm{cm\cdot s^{-1}}$) after 2.5 hours of averaging.
Relative to the $\sim$2.5 GHz line width of the measured transition, this
frequency precision splits the line by a factor of 25,000.

In evaluating the results shown in Figure 2, it is also important to consider
how atmospheric variability (e.g. changes in temperature and pressure) could
influence the observed line shift. To assess the strength of these effects
relative to wind-induced Doppler shifts, we reran the atmospheric model while
including variability in the atmospheric pressure and temperature but
neglecting Doppler effects. For the data on October 12, surface pressure
increased from approximately 834 to 836 hPa over the course of data collection
based on measurements from a co-located weather station. Using our model, we
estimate that an increase in atmospheric pressure at this level could affect a
frequency shift of approximately 80 kHz over the course of the five-hour
measurement with a sign opposite that of the wind-induced shifts. Similarly,
we use the ERA5 data to estimate changes in atmospheric temperature, and we
find that temperature variability induces shifts by $\sim$40 kHz. In both
cases, the pressure- and temperature-induced shifts are small relative to
wind-induced Doppler shifts.

Figure 3: Line-of-sight wind speeds for October 12, 2022 determined using the
ERA5 wind fields. Times are specified relative to the start of data collection
(10:10 a.m. local time, UTC-6).

Furthermore, although our analysis has involved only relative frequency
shifts, it is interesting to consider the use of comb-calibrated LHR to
measure absolute shifts (and thus absolute wind speeds). Past LHR-based wind
measurements have determined vertically-resolved, absolute wind speeds using
inversion methods that rigorously fit the measured spectra with an atmospheric
model [12, 11]. This approach represents a significant increase in complexity
when compared to relative shift measurements, which are only concerned with
deviations from the average. The relative shifts shown in Figure 2 depend only
on the stability of the spectrometer, and a measurement of absolute shifts
would depend on additional factors such as the accuracy of the atmospheric and
spectroscopic data used to fit the measured spectra. Nonetheless, comb-
calibrated LHR may still provide valuable benefits for absolute wind
measurements by leveraging the stability and absolute frequency accuracy of
the comb calibration to reduce instrumental uncertainties and enable precision
tracking of absolute Doppler shifts over long time scales. Future studies
could explore how these benefits impact absolute wind measurements when
combined with a more advanced retrieval procedure.

In conclusion, we demonstrate high-precision spectroscopy of atmospheric CO2
through the unique combination of a laser heterodyne radiometer and an optical
frequency comb. We show that our measurements track wind-induced Doppler
shifts in the measured CO2 spectra with a precision of $\sim$100 kHz
($\mathrm{15\ cm\cdot s^{-1}}$), equivalent to a fractional frequency
precision of a few parts in $10^{10}$. These results demonstrate the potential
of frequency-comb-calibrated LHR as an approach for precision atmospheric
spectroscopy and Doppler metrology. LHR has a long heritage as a technique for
remote sensing of greenhouse gas mixing ratios, and future efforts could seek
to combine these established capabilities with precision Doppler wind
measurements. Such efforts could significantly expand the capabilities of LHR
as a climate monitoring tool and provide valuable data to constrain emissions
estimates and greenhouse gas transport.

More broadly, our results validate comb-calibrated LHR as a tool for precision
Doppler velocimetry that could be of use in applications beyond climate
monitoring. Such applications may include passive tracking of thermal objects
or precision radial velocity measurements of astronomical sources, including
characterizing the impact of telluric absorption on those measurements [16,
24]. In the latter application, achieving precision Doppler spectroscopy at
the $\mathrm{cm\cdot s^{-1}}$ levels represents an ongoing challenge in the
fields of solar and exoplanet science that could be explored in future studies
using comb-calibrated LHR.

Funding This work was supported by the NIST IMS program, NIST financial
assistance award 70NANB18H006, and the NASA Astrophysics Division. R.C.
acknowledges support from the National Academies NRC Research Associateship
Program.

Acknowledgments The authors thank Eugene Tsao and David Plusquellic for
valuable comments and discussions. This work is a contribution of NIST and is
not subject to copyright in the United States. Mention of specific products or
trade names is for technical and scientific information and does not
constitute an endorsement by NIST.

Disclosures The authors declare no conflicts of interest.

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