Video observations of tiny near-Earth objects with Tomo-e Gozen

Abstract

We report the results of video observations of tiny (diameter less than 100 m) near-Earth objects (NEOs) with Tomo-e Gozen on the Kiso 105 cm Schmidt telescope. The rotational period of a tiny asteroid reflects its dynamical history and physical properties since smaller objects are sensitive to the Yarkovsky–O’Keefe–Radzievskii–Paddack (YORP) effect. We carried out video observations of 60 tiny NEOs at 2 fps from 2018 to 2021 and successfully derived the rotational periods and axial ratios of 32 NEOs including 13 fast rotators with rotational periods less than 60 s. The fastest rotator found during our survey is 2020 HS|$_\mathsf {7}$| with a rotational period of 2.99 s. We statistically confirmed that there is a certain number of tiny fast rotators in the NEO population, which have been missed with all previous surveys. We have discovered that the distribution of the tiny NEOs in a diameter and rotational period (D–P) diagram is truncated around a period of 10 s. The truncation with a flat-top shape is not explained well by either a realistic tensile strength of NEOs or the suppression of YORP by meteoroid impacts. We propose that the dependence of the tangential YORP effect on the rotational period potentially explains the observed pattern in the D–P diagram.

1 Introduction

As of 2022 March, 28527 near-Earth objects (NEOs) have been discovered by wide-field monitoring surveys such as the Catalina Sky Survey (CSS; Drake et al. 2009), the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS; Chambers et al. 2016), and the Asteroid Terrestrial-impact Last Alert System (ATLAS; Tonry et al. 2018).¹ Most NEOs have their origins in the main belt (e.g., Bottke et al. 2000; Granvik et al. 2018). Asteroidal fragments are generated from collisional events in the main belt and their orbital elements are gradually changed by the Yarkovsky effect, which is a thermal force caused by radiation from the Sun (e.g., Vokrouhlický 1998; Vokrouhlický et al. 2000; Bottke et al. 2006). When the asteroids enter into orbital resonances with giant bodies, their orbits evolve to those of NEOs in a few Myr (e.g., Gladman et al. 1997; Bottke et al. 2006). During the orbital evolution, the rotational states (i.e., rotational period and pole direction) of the object are changed by the Yarkovsky–O’Keefe–Radzievskii–Paddack (YORP) effect, which arises from the asymmetricity of scattered sunlight and thermal radiation from its surface (e.g., Rubincam 2000; Vokrouhlický & Čapek 2002; Čapek & Vokrouhlický 2004; Bottke et al. 2006). The YORP effect caused by a recoil force normal to the surface, NYORP, is investigated intensively in previous studies. Recently, the tangential YORP (TYORP) effect, which is caused by a recoil force parallel to the surface, was proposed by Golubov and Kruguly (2012).

Since the strength of the YORP effect increases with decreasing diameter, smaller asteroids would experience a larger change in the rotational states. Thus, YORP is a dominant mechanism to change the rotational states of kilometer-sized or smaller asteroids (Vokrouhlický & Čapek 2002). The rotational acceleration by YORP leads to deformation or rotational fission of the asteroid due to a strong centrifugal force. Because the YORP strength is also dependent on physical properties such as shape and thermal conductivity, the rotational period distribution of smaller objects probably reflects the dynamical history and physical properties.

In general, it is difficult to constrain the rotational states of tiny asteroids due to limited observational windows (hours to days), fast rotation (less than a minute), and large apparent motion on the sky (a few arcsec s⁻¹). Observations with exposure times sufficiently shorter than their rotational periods are required. The shorter exposure times are effective in suppressing the trailing sensitivity loss effect, which is the degradation of the surface brightness of a moving object on an image (Zhai et al. 2014).

The Asteroid Lightcurve Database (LCDB; Warner et al. 2009) has thousands of rotational periods of minor planets. The diameter and rotational period relation (hereinafter referred to as the D–P relation) is shown in figure 1. As of 2021 June, rotational periods of 5060 objects are estimated with high accuracy [the quality code U in Warner et al. (2009) is 3 or 3-]. For asteroids larger than 200 m in diameter, the rotation period distribution is truncated at around two hours. This clear structure is called the cohesionless spin barrier and indicates that most of the larger asteroids are rubble-piles (Pravec & Harris 2000). It is possible to constrain physical properties of asteroids smaller than 200 m in diameter from the D–P relation in the same way as the larger asteroids. However, there is a smaller number of asteroids for which the rotational period has been reported so far.

The LCDB contains the observational results of the Mission Accessible Near-Earth Objects Survey (MANOS), which obtained more than 300 light curves of small NEOs with the mean absolute magnitude of about 24 mag using large and medium aperture telescopes (Thirouin et al. 2016, 2018). Although MANOS successfully derived the rotational periods of NEOs with high accuracy, the main motivation of the survey is not to detect fast rotators, but to characterize mission-accessible NEOs. Due to a relatively long exposure time (1–300 s), the survey possibly did not detect the very fast rotations. Systematic high-speed observations are required to derive shorter rotational periods correctly and obtain an unbiased D–P relation of tiny NEOs.

In this paper, we report the results of imaging observations at 2 fps of 60 tiny NEOs with the wide-field CMOS camera Tomo-e Gozen. The observed NEOs are smaller than 100 m in diameter and their mean diameter is 20 m. The aims of this study are to obtain an unbiased D–P relation by video observations and to reveal dynamical histories and physical properties of tiny NEOs. Observations and data reduction are described in section 2. The results are compared with previous studies in section 3. In section 4, the D–P relation of the tiny NEOs obtained in this study is discussed taking into account the spin acceleration by YORP.

2 Observations and data reduction

2.1 Observations

We conducted photometric observations at 2 fps with the wide-field CMOS camera Tomo-e Gozen (Sako et al. 2018). Tomo-e Gozen is a wide-field high-speed camera mounted on the 105 cm Schmidt telescope at Kiso Observatory (Minor Planet Center code 381) in Nagano, Japan. The field of view is 20.7 square degrees covered by 84 chips of CMOS sensors without photometric filters. A timestamp of each image data is GPS-synchronized and has a time accuracy of 0.2 ms. We have performed 2 fps all-sky survey observations with Tomo-e Gozen since 2019. Data accumulated each night amount to 30 TB, from which various types of transients such as supernovae and tiny NEOs are searched for. Tomo-e Gozen has discovered 32 NEOs from the survey data in real time from 2019 March to 2021 October with the fast-moving object pipeline using a machine-learning technique (R. Ohsawa et al. in preparation). The algorithms used in the pipeline are partly described in Ohsawa (2021).

In this paper, we assume that p_V is 0.20, which is a typical value for S-type asteroids, as used in LCDB. The absolute magnitude of 22.5 corresponds to 94 m in diameter. Since a median rotational period of NEOs in LCDB satisfying the quality code U of 3 or 3- and H smaller than 22.5 is about 9 min, we set the nominal duration of observation at 20 min. The mean absolute magnitude of our samples is 26.6, corresponding to 14 m in diameter. As shown in figure 2, the peak of the distribution (H ∼ 26) is smaller than the peak of the targets observed by MANOS (H ∼ 24), hereinafter referred to as the MANOS NEOs.

The Tomo-e NEOs were typically located at a few lunar distances from Earth when observed. A typical angular velocity was about a few arcsec s⁻¹. Most of the Tomo-e NEOs were discovered a few hours or a few days before our observations, except for 2010 WC₉, 2011 DW, and 2017 WJ₁₆. TMG0042 and TMG0049 are NEO candidates discovered by Tomo-e Gozen. Due to the limited number of follow-up observations, provisional designations for the two objects were not taken from the Minor Planet Center.

To obtain the light curve of the NEO, we used a single sensor of Tomo-e Gozen with a field of view of |${39{^{\prime }_{.}}7} \times {22{^{\prime }_{.}}4}$| and a pixel scale of |${1{^{\prime \prime}_{.}}189}$|⁠. Sidereal tracking and re-pointing were performed to follow the fast-moving NEOs. All of the Tomo-e NEOs except for 2018 LV₃ were observed at 2 fps. The light curve of 2018 LV₃ was obtained at 0.2 fps as an experimental observation.

2.2 Data reduction

2.2.1 Photometry

Observations are composed of a series of video data which were typically 1 min in length. The video data were compiled into cube FITS files. After bias and dark subtraction and flat-field correction, standard circle aperture photometry was performed on a target and reference stars in each frame using the SExtractor-based python package sep (Bertin & Arnouts 1996; K. Barbary et al. 2015⁴). Since the elongations of the NEOs were negligible, we applied the standard aperture photometry method. The aperture radius was set to two to three times larger than the full width at half maximum (FWHM) of the point spread function (PSF) of reference stars, which was typically 3″ to 5″. We determined the FWHM of the stellar PSF in the first frame of the cube, and then conducted the photometry of the objects in each frame. Sometimes the target was too faint to be detected, possibly because of the brightness variation of the target. In such cases, we set the aperture at the expected positions interpolated from the positions in adjacent frames and performed forced photometry.

We used the G-band magnitude from the Gaia DR2 catalog as brightness references since the spectral response of Tomo-e Gozen (350 to 950 nm; Kojima et al. 2018) is similar to that of the G-band of Gaia (330 to 1050 nm; Gaia Collaboration 2018). The difference in the spectral responses may affect the mean apparent magnitudes of the NEOs, but the rotational period is not affected by the spectral response and the effect on the amplitudes is negligible. The discussion in this paper is not affected.

The observed G-band magnitudes were converted to reduced magnitudes with the distance between the Sun and NEO (r) and NEO and observer (Δ) at the time of observations. The phase angle correction and the light-travel time correction were done to obtain the corrected light curves.

2.2.2 Periodic analysis

The uncertainty of the rotational period and the light-curve amplitude were estimated using the Monte Carlo method. We created 3000 light curves for each NEO by randomly resampling the data assuming each observed data follows a normal distribution the standard deviation of which is a photometric error. We performed the same analyses above for 3000 light curves and obtained 3000 sets of Fourier coefficients in equation (3). We calculated the 3000 periods and the 3000 amplitude with the corresponding peak frequencies for each light curve. As an example, the analysis result of 2021 CG is shown in figure 3. We adopted the standard deviations as the uncertainties of the period and the amplitude. If the estimated standard deviation was larger than |$5\%$| of the rotational period, we judged that the derived rotational period was suspicious and the correct rotational period was not derived. We used the Fourier coefficients of which the rotational period and the light curve amplitude are the closest to the average values to plot a typical model curve.

There are several candidates of non-principal axis rotators (i.e., tumbler) in the Tomo-e NEOs. The tumbler is in a excited state and its light curve is complicated (e.g., Paolicchi et al. 2002; Pravec et al. 2005). Therefore, the periodograms of the tumblers show additional peaks which are not aliases of the highest peak. We defined such objects with multiple peaks as tumbler candidates in this paper.

3 Results

We successfully derived the rotational periods of 32 NEOs. Results of the analysis are summarized in table 2. The rotational periods of 11 out of the 32 are reported in previous studies: 2019 BE₅ (Warner et al. 2009), 2020 TD₈, 2020 UQ₆, 2020 VZ₆, 2020 XX₃ (Birtwhistle 2021a), 2021 EX₁, 2021 FH (Birtwhistle 2021b), 2021 KN₂, 2021 JB₆, 2021 GQ₁₀ (Birtwhistle 2021c), and 2021 DW₁ (Kwiatkowski et al. 2021). All our results are consistent with the reported values. The periodogram of 2021 FH has a prominent peak but its significance level is lower than |$99.9\%$|⁠. We considered that the peak of 2021 FH is reliable since the peak frequency corresponds to the rotational period (63.4 s) reported by Birtwhistle (2021b).

The rotational periods of 18 objects were not derived due to small amplitudes. These objects may have axisymmetric shapes, rotational periods longer than the duration of observation, rotational periods shorter than the exposure time, or rotational axes parallel to the line of sight. When a light curve shows a clear brightness variation but whole cycles of rotation were not obtained, we adopted the duration times of the observations as lower limits of rotational periods.

We found 10 tumbler candidates: 2010 WC₉, 2017 WJ₁₆, 2020 TE₆, 2021 CO, 2020 EO, 2021 KN₂, 2021 KQ₂, 2021 TG₁, 2021 TL₁₄, and 2021 TQ₄. Physical modeling of these candidates will be presented elsewhere.

3.1 Light curves and periodograms

As an example, we presented the light curve and periodogram of 2021 CG in figures 4 and 5, respectively. The rotational period and the light curve amplitude of 2021 CG were estimated to be 15.296 ± 0.002 s and 0.27 ± 0.02 mag, respectively, using the Monte Carlo method as shown in figure 3. The light curve folded by the rotational period (hereinafter referred to as the phased light curve) is shown in figure 6. Thanks to the video observations at 2 fps, we can estimate such a short period of rotation with high reliability. (The light curves, periodograms of 60 NEOs, and phased light curves of NEOs whose rotational periods were estimated are presented in figures 14–16 in the Appendix.)

3.2 D–P relation

The D–P relation of the Tomo-e NEOs and the NEOs in LCDB is shown in figure 7. The Tomo-e NEOs are distributed in a range of 3 to 100 m in diameter and 3 to 420 s in rotational period. We found 13 NEOs with rotational periods less than 60 s.

We create cumulative histograms of rotational periods of the Tomo-e NEOs and the NEOs in LCDB (figure 8). The D–P relation of the Tomo-e NEOs looks different from that of the NEOs in LCDB. We performed the Kolmogorov–Smirnov (KS) test to check the null hypothesis that the two D–P relations are the same. We chose the NEOs satisfying the criteria that the absolute magnitude is larger than 22.5 and the rotational period is shorter than 420 s corresponding to the longest rotational period of the Tomo-e NEOs. The NEOs which had quality codes 3 or 3- are used as for the NEOs in LCDB. The deduced KS statistics and the p-value are 0.330 and 0.013, respectively. This tentatively implies that rotational periods of some fast rotators could not be estimated due to long exposure times and other factors in the previous studies.

3.3 Axial ratios

The relation between the absolute magnitudes H and the lower limits of axial ratios a/b of the Tomo-e NEOs and the MANOS NEOs with rotational periods shorter than 600 s are shown in figure 9. The relation between the rotational period P and a/b of the Tomo-e NEOs and the MANOS NEOs with P ≤ 600 s are shown in figure 10. The mean of a/b for each range of H and P is also presented. The range is determined based on the Sturges’ rule. No strong correlation is seen in either figure 9 or 10. The present results are consistent with those of Hatch and Wiegert (2015) and Thirouin et al. (2016).

The difference of mean axial ratios between the Tomo-e NEOs (∼1.29) and the MANOS NEOs (∼1.27) is about 0.02. We performed a bootstrap test to check the null hypothesis that the mean axial ratios of the two samples are the same. We generated 10000 differences of the mean axial ratios by resampling the Tomo-e and MANOS NEOs. The |$95\%$| confidence interval is from −0.08 to 0.14. Thus, the null hypothesis is not rejected at the |$5\%$| significance level.

Figure 11 shows measured a/b of various sources: the average of the sum of the Tomo-e NEOs and the MANOS NEOs, the average of fast-rotating asteroids (FRAs) with diameters less than 200 m and a rotational period less than 1 hr (Michikami et al. 2010), and the averages of boulders on the surfaces of asteroids Itokawa (Michikami et al. 2010) and Ryugu (Michikami et al. 2019).

Michikami et al. (2010) mention that the lower limits of a/b of FRAs and the a/b of boulders are similar to those of laboratory experiments (∼1.4), although the aspect angles of asteroids are unknown. The lower limits of a/b of asteroids are lower in the case of recent observational results such as Tomo-e Gozen (∼1.29) and MANOS (∼1.27). It is important not only to increase the number of light curve observations, but also to determine pole directions to discuss the relation with fragments of laboratory experiments and boulders (e.g., Kwiatkowski et al. 2021).

4 Discussion

4.1 Detectable rotational period

Sparse sampling and finite exposure time may lead to underestimating the light curve amplitude and misidentifying periodicity (Pravec & Harris 2000; Thirouin et al. 2018; Birtwhistle 2021c). We examine detection limits in rotational periods (P_det) in our observations to verify the deficiency of asteroids rotating faster than 10 s. We simulate light curves if an asteroid was rotating faster than it is, and then the light curves are analyzed in the same manner. We selected eight NEOs with rotational periods that are short (P ≤ 60 s), and the durations of the observations are typical (10 ≤ T ≤ 30 min). We excluded 2020 HS₇ from this examination since its exposure time is not sufficiently shorter than the rotational period and the observed light curve could be underestimated (Birtwhistle 2021c). The model light curves of the eight NEOs are used as templates. The rotational period of a hypothetical asteroid P_pseudo is set to P_pseudo = P/2, P/3, P/4, ..., where P is the original rotational period in section 3. Then, the hypothetical asteroid is virtually observed to generate a pseudo light curve. The number of measurements (N_obs) and the timestamps are the same as the actual observation. The pseudo light curve is perturbed to match the noise level with the original observation. The criterion of the periodicity identification is the same as in sub-subsection 2.2.2.

The results of the periodic analysis are summarized in table 3. The detectable rotational periods are less than 2 s for all the eight asteroids. Although our observations are unevenly sampled because of intervals between frames, large fractions of the data are evenly sampled at 2 fps. The periodograms of the pseudo light curves can be affected by aliases when the frequency gets closer to the Nyquist limit f_Ny = 2/2 = 1 Hz. However, the peaks by the aliases become weaker than the genuine peaks due to the uneven sampling. Thus, it is natural that we detect shorter rotational periods than the Nyquist limit (P = 2 s assuming typical double-peak light curves). We conservatively set the detectable rotational period to 1.5 s in our systematic 20 min video observations at 2 fps. Therefore, it is inevitable that there is only one fast rotator whose rotational period is shorter than 10 s from our 60 NEOs.

4.2 Deficiency of fast rotators

We found no NEOs with rotational periods shorter than 10 s other than 2020 HS₇. The distribution of the Tomo-e NEOs in the D–P relation is truncated around 10 s in the rotational period, as shown in figure 7. To interpret this flat-top distribution, we consider the evolution of rotational periods of the NEOs.

Since smaller asteroids experience a stronger Yarkovsky effect and their semi-major axes are changed, parts of them drift to the resonances with giant planets in the main belt in a short time scale (∼a few Myr) and then are scattered into the near-Earth region (Bottke et al. 2006). The orbits of the scattered asteroids evolve to those of NEOs over a few Myr (Gladman et al. 1997). Therefore, typical NEOs are considered to be a few to 10 Myr old. This timescale (hereinafter referred to as NEO age) is consistent with the typical cosmic ray exposure age of meteorites (Eugster et al. 2006).

Since YORP gradually changes the rotational states of NEOs during the orbital evolution, the distribution of the rotational periods reflects the NEO age. Although YORP also decelerates the rotation, here we consider only the acceleration. The decelerated tiny asteroids shortly enter tumbling states once spinning down starts (Vokrouhlický et al. 2007; Breiter et al. 2011) and it is difficult to predict their evolution accurately. In this study, we estimate reachable rotational periods of NEOs by the YORP acceleration. For the sake of the simplicity, we do not take into account the time evolution of the orbital elements, resulting in a constant acceleration.

Figure 12 shows isochrones for t = 0.1, 1, 10, 100, and 1000 Myr. Based on the isochrones, tiny NEOs with diameters less than 10 m and ages older than 10 Myr, corresponding to the typical dynamical evolution timescale of the NEOs, rotates faster than about 10 s. However, no such fast rotators are found, other than 2020 HS₇. The observed truncation is not produced by the constant acceleration model.

The densities and surface properties of NEOs depend on their sizes (Carry 2012). However, the density difference of NEOs is a factor of a few at most and does not suppress the acceleration of rotation sufficiently. The thermal inertia also has little effect on the rotational period (Čapek & Vokrouhlický 2004; Golubov et al. 2021). Therefore, other dynamical mechanisms are required to explain the flat-top distribution.

4.2.1 Tensile strength

We present two lines indicating P_cri with tensile strength of typical stony meteorites (κ = 10⁵ N m^−3/2, Kwiatkowski et al. 2010) and weak material (κ = 10³ N m^−3/2), respectively, in panel (b) of figure 12. We use a typical density of S-type asteroids (ρ = 2500 kg m⁻³). In the case of weak material, we can explain the deficiency of NEOs with D ≤ 10 m and P ≤ 10 s. However, the flat-top shape of the distribution is not reproduced because P_cri is proportional to D^5/4.

4.2.2 Suppression of YORP by meteoroid impacts

The YORP acceleration can be suppressed by meteoroid impacts on to an asteroid surface (Farinella et al. 1998; Wiegert 2015). We investigate the evolution of the rotational period taking into account possible effects by meteoroid impacts. We discuss two effects related to meteoroid impacts: angular momentum transfer and cratering.

4.2.2.1 Angular momentum transfer

When a collision with a large δL/L occurs, the spin axis of the asteroid can be tilted, leading to a ceasing of the YORP acceleration. Therefore, a timescale of such a critical collision, τ_L, corresponds to the duration of the YORP acceleration,

We adopt a β of 20, v_imp of 3 × 10⁴ m s⁻¹, and ρ of 2500 kg m⁻³ as typical quantities. The timescale τ_L provides the possible fastest rotational period accelerated by YORP. We present three limiting lines for different δL/L values in panel (c) of figure 12. The YORP acceleration of smaller asteroids is more suppressed by the angular momentum transfer. However, the flat-top shape of the distribution is not reproduced because the reachable periods are proportional to D.

4.2.2.2 Cratering

where m₀ and m₁ are minimum and maximum masses of the meteoroids, respectively. We set |$m_1 \rightarrow \inf$| and m₀ as a free parameter.

We adopt a ρ of 2500 kg m⁻³ and v_imp of 3 × 10⁴ m s⁻¹. The possible fastest rotational periods are presented over a wide range of m₀ values in panel (d) of figure 12. The possible fastest rotational period is approximately proportional to D. Therefore, we cannot explain the truncated distribution with the suppression of YORP by cratering.

4.2.3 Tangential YORP effect

We have considered only the normal YORP (NYORP), disregarding tangential YORP (TYORP). TYORP depends on the rotational period and thermal properties of the asteroid, as with NYORP. In most cases, TYORP contributes to the acceleration of the rotation unlike NYORP, which decelerates the rotation as well (Golubov & Kruguly 2012; Golubov et al. 2014).

5 Conclusions

The rotational period of an asteroid reflects its dynamical history and physical properties. We have obtained the light curves of 60 tiny (diameter less than 100 m) NEOs with the wide-field CMOS camera Tomo-e Gozen. We successfully derived the rotational periods and axial ratios of 32 samples owing to the video observations at 2 fps. We found 13 objects with rotational periods less than 60 s. Compared with the literature, the distribution of the rotational periods of 32 objects shows a potential excess in shorter periods. This result suggests that previous studies missed some population of fast-rotating asteroids due to long exposure time observations.

We discovered that the distribution of the tiny NEOs in the D–P diagram is truncated around a period of 10 s. We performed model calculations taking into account the YORP effect. A NEO smaller than 10 m is expected to rotate with a period shorter than 10 s assuming a constant acceleration by YORP, which is not consistent with the present results. The truncated distribution is not well explained by either the realistic tensile strength of NEOs or the suppression of YORP by meteoroid impacts. We found that the tangential YORP effect is a possible mechanism to produce the truncated distribution, although further observational and theoretical studies as well as high-speed light curve observations of NEOs are necessary to reach the conclusion.

Acknowledgements

We give special thanks to Mr. Yuto Kojima for his technical assistance with this study. We would like to thank near-Earth asteroid observers around the world. J.B. would like to express the gratitude to the Iwadare Scholarship Foundation and the Public Trust Iwai Hisao Memorial Tokyo Scholarship Fund for the grants. This work has been supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI grants, 21H04491, 20H04617, 18H05223, 18H01272, 18H01261, 18K13599, 17H06363, 16H06341, 16H02158, 26247074, and 25103502. This work is supported in part by the Optical and Near-Infrared Astronomy Inter-University Cooperation Program, the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, JST SPRING, Grant Number JPMJSP2108, and the UTEC UTokyo Scholarship. This work has made use of data from the European Space Agency (ESA) mission Gaia 〈https://www.cosmos.esa.int/gaia〉, processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

Appendix. Light curves, periodograms, and phased light curves

Light curves, Lomb–Scargle periodograms, and phased light curves of the Tomo-e NEOs are presented here in figures 14–16.

Footnotes

References