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A&A
Volume 661, May 2022
Article Number L3
Number of page(s) 14
Section Letters to the Editor
DOI https://doi.org/10.1051/0004-6361/202243442
Published online 12 May 2022

© ESO 2022

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). It was discovered on June 19, 2004, by R. A. Tucker, D. J. Tholen, and F. Bernardi at Kitt Peak, Arizona. From early predictions, it was estimated that this asteroid could impact Earth with a maximum probability of 2.7% (Jet Propulsion Laboratory 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 it would pass by Earth in April 2029 at a geocentric distance of 38 326 km (approximately six Earth radii), which is within the geosynchronous orbit in April 2029 (Giorgini et al. 2008). Recently, radar observations made in March 2021 at the Goldstone Solar System Radar and the Green Bank Telescope were used to precisely estimate Apophis’s orbit around the Sun, ruling out any Earth impact threat for the next hundred years or more (Greicius 2021).

Although Apophis’s impact threat has disappeared, this asteroid remains an object of interest because of its close approach in 2029. The 2029 Earth encounter 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 is 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 by Pravec et al. (2014) using photometric data obtained from the 2012−2013 apparition. They found it has non-principal axis rotation in a short-axis mode (SAM), with rotation and precession periods of 263 and 27.38 h, respectively, and an orientation of angular momentum vector of λL = 250° and βL = −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 to intermediate principal moments of inertia (Ib/Ic) of 0.965 and a ratio of the greatest to smallest principal moment of inertia (Ia/Ic) 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 Sect. 2. In Sect. 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 Sect. 4.

2. Photometric observation campaign during the 2020–2021 apparition

As mentioned above, the 2020−2021 apparition of Apophis was the last opportunity to study 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 using both the ground-based telescopes and the Transiting Exoplanet Survey Satellite (TESS) space telescope (Ricker et al. 2015). The details of the telescopes and instruments used in our campaign are provided in Table 1. The geometries and observational circumstances are listed in Table A.1. This campaign’s data also contributed to the global Apophis Planetary Defense Campaign (Reddy et al. 2022).

Table 1.

Details of the observatories and instruments used in this campaign.

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 Sloan Digital Sky Survey r′ filter. All data reductions were conducted following standard procedures. However as the observations were made with different telescopes and instruments, the data reduction processes may differ slightly. The bias- and flat-field images were corrected, and the instrument magnitudes for each frame were measured using aperture photometry. All photometric data were calibrated with the ATLAS All-Sky Stellar Reference Catalog (ATLAS Refcat2; Tonry et al. 2018a). The ATLAS Refcat2 magnitudes were converted to Johnson–Cousins V and R magnitudes using empirical transformation equations (Tonry et al. 2012). 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 long-term continuous photometric data observed from the TESS spacecraft using a wide I passband (see Fig. 1 in Ricker et al. 2015). TESS photometric data were obtained in a similar manner as done in Pál et al. (2020) for the image series of Sector 35 acquired between February 19, 2021, and March 7, 2021. 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 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 February 7.0, 2021, to March 16.2, 2021. 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′−R = 0.33. The data were converted to flux units, the heliocentric and geocentric distances were corrected to the unit distance, and the solar phase angle was converted to a consistent value using the H − G phase relation and assuming G = 0.24. As Apophis exhibited tumbling motion, we attempted to detect double periods from the asteroid’s light curve.

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 h. In accordance with this method, we carried out a periodic analysis based on a single-peak light curve. However, most asteroid light curves have double peaks because their shapes are elongated. Therefore, we determined the primary period (P1) on the light curve of Apophis to be 30.56 h, which is double the strongest signal in the Lomb-Scargle periodogram.

The secondary period (P2) 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:

F ( t ) = C 0 + ∑ j = 1 m [ C j 0 cos 2 π j P 1 t + S j 0 sin 2 π j P 1 t ] + ∑ k = 1 m ∑ j = − m m [ C jk cos ( 2 π j P 1 + 2 π k P 2 ) t + S jk sin ( 2 π j P 1 + 2 π k P 2 ) t ] · $$ \begin{aligned} F(t) =&C_{0} + \sum \limits _{j=1}^{m}\Bigg [C_{j0} \cos {\frac{2 \pi j}{P_{1}}}t + {S\!}_{j0} \sin {\frac{2 \pi j}{P_{1}}}t\Bigg ] \\& + \sum \limits _{k=1}^{m}\sum \limits _{j=-m}^{m}\Bigg [C_{jk} \cos {\left(\frac{2 \pi j}{P_{1}} + \frac{2 \pi k}{P_{2}}\right)}t + {S\!}_{jk} \sin {\left(\frac{2 \pi j}{P_{1}} + \frac{2 \pi k}{P_{2}}\right)}t\Bigg ]\cdot \end{aligned} $$

The result for the P2 search is shown in Fig. 1, where the abscissa is P2 and the ordinate is the sum of the squared residuals for the fitted fourth-order two-period Fourier series with P1 = 30.56 h. From our P2 search, we found two minimums, at 27.38 and 263 h. As the long period of 263 h was a combination of the short period of 27.38 h with P1, that is, 263−1 ≈ 27.38−1 − 30.56−1, it seems to be derived from the short period. Thus, we determined P2 to be 27.38 h. Figure 2 shows a composite light curve of Apophis obtained from February 7.0, 2021, to March 16.2, 2021 with the fitted fourth-order two-period Fourier series for periods of 30.56 and 27.38 h.

thumbnail Fig. 1.

P2 search diagram of Apophis. The sum of square residuals was calculated for the fourth-order two-period Fourier series with P1 = 30.56 h fitted to our dense light curve obtained from February 7.0, 2021, to March 16.2, 2021 in flux units.

thumbnail Fig. 2.

Light curve of Apophis taken from February 7.0, 2021, to March 16.2, 2021 reduced to the unit geocentric and heliocentric distance and to a consistent solar phase angle. The blue open circles indicate the photometric data folded with P1. The red curves denote the best-fit fourth-order two-period Fourier series with the periods P1 = 30.56 h and P2 = 27.38 h. The black squares indicate the residuals of the photometric data from the fourth-order two-period Fourier series (see the right ordinates).

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

Given that the radar observation of Apophis suggests 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 the data at the phase angles needed 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 non-principal axis rotator (Kaasalainen & Torppa 2001; Kaasalainen et al. 2001; Kaasalainen 2001) combined with Hapke’s light-scattering model (Hapke 1993). In this work, we used both the light curves obtained through our campaign observation and all available observation data collected from the literature. The historical light curves are listed in Table A.2.

The first step in revealing the spin state of a tumbling asteroid is to determine the physical periods, that is, the rotation (Pψ) and precession period (Pϕ). Because the periods on the light curve were derived from the physical periods, P 1 − 1 $ P_{1}^{-1} $ and P 2 − 1 $ P_{2}^{-1} $ usually appear at P ϕ − 1 $ P_{\phi}^{-1} $ and P ϕ − 1 ± P ψ − 1 $ P_{\phi}^{-1} \pm P_{\psi}^{-1} $, where the plus sign is for the long-axis mode (LAM) and the minus sign for the SAM (Kaasalainen 2001). Therefore, we found two possible physical period combinations using the periods obtained in the previous section: P 1 − 1 = P ϕ − 1 $ P_{1}^{-1} = P_{\phi}^{-1} $ and P 2 − 1 = P ϕ − 1 + P ψ − 1 $ P_{2}^{-1} = P_{\phi}^{-1} + P_{\psi}^{-1} $ (LAM); P 1 − 1 = P ϕ − 1 − P ψ − 1 $ P_{1}^{-1} = P_{\phi}^{-1} - P_{\psi}^{-1} $ and P 2 − 1 = P ϕ − 1 $ P_{2}^{-1} = P_{\phi}^{-1} $ (SAM). The optimization for each Pϕ was performed in the same way as in Lee et al. (2020, 2021). In these procedure, the Hapke’s model parameters were set to those of a typical S-type asteroid: ϖ = 0.23, g = −0.27, h = 0.08, B0 = 1.6, and θ ¯ = 20 ° $ \bar{\theta} = 20^\circ $ (Li et al. 2015). Because we did not obtain data observed at low solar phase angles, the parameters for the opposition surge, h and B0, and the roughness, θ ¯ $ \bar{\theta} $, were fixed. Only the ϖ 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 I1 and I2 parameters. Therefore, we decided to use the SAM as the final solution.

The best-fit model for Apophis is listed in Table 2, together with the Pravec et al. (2014) solution for comparison. The uncertainties of our model parameters correspond to a 3σ confidence interval. The confidence interval was estimated from the increase in the χ2 value when the solved-for physical parameters were varied. The threshold corresponding to the 3σ confidence interval was set as χ min 2 × ( 1 + 3 2 / ν ) $ \chi^{2}_{\mathrm{min}} \times (1 + 3 \sqrt{2/\nu}) $1, where χ min 2 $ \chi^{2}_{\mathrm{min}} $ represents the χ2 value for the best-fit solution and ν represents the number of degrees of freedom (Vokrouhlický et al. 2017). The convex shape model of Apophis is shown in Fig. 3. The synthetic light curve of our model and the observed light curve are presented in Appendix B.

thumbnail Fig. 3.

Convex shape model of Apophis.

Table 2.

Parameters of the Apophis model.

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 ground-based telescopes and TESS. We obtained 211 dense-in-time light curves and two spare-in-time light curves. In the period analysis conducted with our dense light curves, double periods of 30.56 and 27.38 h, respectively, were detected.

The best-fit solution indicates that Apophis is in a SAM state with rotation and precession periods of 264.178 ± 0.01 and 27.38547 ± 0.00002 h, respectively. The ecliptic coordinates of the angular momentum vector orientation of this asteroid are (275 ± 3°, −85 ± 1°). In addition, the ratios of the dynamical moments of inertia were estimated to be Ia/Ic = 0.64 and Ib/Ic = 0.96. Our model is similar to that of Pravec et al. (2014). Nonetheless, the uncertainties of the model parameters were improved because they were reconstructed based on the data set obtained from the two apparitions. This model will be useful both for investigating changes in Apophis’s physical properties due to the tidal effect during its encounter in 2029 and for planning a space mission to this asteroid.

1

These was a typo in Vokrouhlický et al. (2017), so we used 2/ν instead of 2ν.

Acknowledgments

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 Letter 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; by products 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. M.K. and O.E. thank TUBITAK National Observatory for a partial support in using T100 telescope with project number 20CT100-1743.

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Appendix A: Additional tables

Table A.1.

Geometries and observational circumstances.

Table A.2.

List of the historical light curves.

Appendix B: Light curves

thumbnail Fig. B.1.

Photometric data from 2012-2013 (black open circle) with the synthetic light curve from the best-fit model (red line).

thumbnail Fig. B.2.

Photometric data from 2020-2021 (black open circle) with the synthetic light curve from the best-fit model (red line).

All Tables

Table 1.

Details of the observatories and instruments used in this campaign.

In the text
Table 2.

Parameters of the Apophis model.

In the text
Table A.1.

Geometries and observational circumstances.

In the text
Table A.2.

List of the historical light curves.

In the text

All Figures

thumbnail Fig. 1.

P2 search diagram of Apophis. The sum of square residuals was calculated for the fourth-order two-period Fourier series with P1 = 30.56 h fitted to our dense light curve obtained from February 7.0, 2021, to March 16.2, 2021 in flux units.
In the text
thumbnail Fig. 2.

Light curve of Apophis taken from February 7.0, 2021, to March 16.2, 2021 reduced to the unit geocentric and heliocentric distance and to a consistent solar phase angle. The blue open circles indicate the photometric data folded with P1. The red curves denote the best-fit fourth-order two-period Fourier series with the periods P1 = 30.56 h and P2 = 27.38 h. The black squares indicate the residuals of the photometric data from the fourth-order two-period Fourier series (see the right ordinates).
In the text
thumbnail Fig. 3.

Convex shape model of Apophis.
In the text
thumbnail Fig. B.1.

Photometric data from 2012-2013 (black open circle) with the synthetic light curve from the best-fit model (red line).
In the text
thumbnail Fig. B.2.

Photometric data from 2020-2021 (black open circle) with the synthetic light curve from the best-fit model (red line).
In the text

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