Asteroid Discovery and Light Curve Extraction Using the Hough Transform: A Rotation Period Study for Subkilometer Main-belt Asteroids

Finding asteroids is the fundamental step for asteroid research. With the advances in wide-field cameras, robotic observations, and computing resources, several telescopes have been dedicated to asteroid discovery, such as the Catalina Sky Survey (CSS),⁶ the Pan-STARRS 1 (Chambers et al. 2016), the Asteroid Terrestrial-impact Last Alert System (ATLAS; Tonry et al. 2018), and the Chinese Near-Earth-Object Survey Telescope (CNEOST).⁷ To optimize the discovery rate of asteroids, each project customizes its survey rates (i.e., scanning the sky area over a period of observation time) to a certain cadence that meets the basic requirement of discovering an asteroid, such as three detections within a night or several detections over a few nights for a single asteroid discovery. Therefore, each project develops its own asteroid detecting pipeline and optimizes it to the scientific interests and survey strategy. In some cases, the pipeline does not utilize all of the detections for some reason. The most common case is a filter to exclude the low signal-to-noise ratio detections, which usually contain many noises. Nevertheless, if a survey was conducted using a relatively short cadence, asteroids could be distinguished much easier from the background noises and field stars. A good example of high-cadence observation is the rotation period survey for main-belt asteroids (MBAs) carried out in 2016 October using the Pan-STARRS 1 (PS1; Chang et al. 2019), in which eight PS1 fields were continuously scanned using a ∼10 minute cadence over six straight nights. From the survey, the intranight detections of asteroids appeared as straight lines due to their approximately linear motion during the short period of time, which obviously stand out from the stationary sources and noises. Therefore, we can make use of the Hough transform to connect these intranight line-up detections to find asteroids. The Hough transform was originally proposed by Hough (1959) and then further utilized by Duda & Hart (1972; i.e., generalized Hough transform). This method has been widely used to extract the linear features on two-dimensional images and recently started to be applied to astronomical images to detect cosmic rays (Keys & Pevtsov 2010), streaks produced from fast-moving small solar system bodies (SSSBs; Virtanen et al. 2014; Kim 2016; Bektešević & Vinković 2017; Nir et al. 2018), and long tracks of space debris (Hickson 2018). Instead of working on images, we applied the Hough transform to the source catalogs obtained from the aforementioned intranight observations. The implementation of the Hough transform is relatively simple and, moreover, it does not have to exclude low signal-to-noise detections in advance. Therefore, we were able to utilize the data set much closer to its actual limiting magnitude (i.e., ∼22.5 mag) and found more than 3000 asteroids mainly of 21.5 < w_p1 < 22.5 mag. With this magnitude, a study of rotation period for subkilometer MBAs was carried out for the first time.⁸

In this work, the PS1 observation and the data set are briefly described in Section 2, the Hough transform is illustrated in Section 3, the result of asteroid recovery, discovery, and rotation period analysis are presented in Section 4, and a short summary is given in the end.

PS1 was designed to discover small solar system bodies and, especially, those potentially hazardous objects. It is a 1.8 m Ritchey–Chretien reflector located on Haleakala, Maui equipped with the gigapixel camera having a field of view of 7 deg². Six filters are used, including g_P1 (∼400–550 nm), r_P1 (∼550–700 nm), i_P1 (∼690–820 nm), z_P1 (∼820–920 nm), y_P1 (>920 nm), and w_P1 (i.e., combination of g_P1, r_P1, and i_P1), in which the w_P1 filter is especially designed for moving object discovery (Kaiser et al. 2010; Tonry et al. 2012; Chambers et al. 2016). The images obtained by PS1 are processed by the Image Processing Pipeline (IPP; Chambers et al. 2016; Magnier et al. 2016a, 2016b, 2016c; Waters et al. 2016) to provide source catalogs as the final product.

During 2016 October 26–31, a high-cadence observation was carried out using PS1 to collect a large sample of asteroid light curves for a rotation period measurement (Chang et al. 2019). The observation was conducted using a cadence of ∼10 minutes to repeatedly scan eight consecutive PS1 fields in the w_P1 band. Therefore, each field had ∼30 exposures each night, except for the last two nights which only had a few exposures that were taken due to unstable weather. Moreover, several exposures in each g_P1, r_P1, i_P1, and z_P1 band were also taken in the first night between the exposures of the w_P1 band to obtain asteroid colors. Table 1 shows the summary of the observation. Although ∼3500 objects, mostly <21.5 mag, have been discovered and reported to the Minor Planet Center⁹ using this observation, a lot of unknown objects close to the actual limiting magnitude of ∼22.5 have not been found yet. This high-cadence observation is suitable for asteroid discovery using the Hough transform. Therefore, the source catalogs obtained from the observation were used as our running cases.

3.1. The Hough Transform

The intranight trajectory of an asteroid can be approximated by a straight line, and so do its detections are obtained from intranight observations. Therefore, we can find asteroids through these line-up intranight detections. The situation is illustrated in Figures 1 and 2, in which all of the detections, obtained from all the source catalogs of the intranight observations for one particular field, are stacked into a single frame (hereafter, the master frame). On the master frame we see that the intranight detections of asteroids on that field appear as straight lines with correct time sequence.

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Using the Hough transform, a line on a two-dimensional image can be expressed in the Hesse normal form as

$$\begin{eqnarray}&&r=x\cos \theta +y\sin \theta ,\end{eqnarray} \tag{ 1 }$$

where (x, y) is the coordinate of the pixel on the image, r is the distance to the closet point of the line to the reference point, and θ is the angle between this line and the x-axis (see Figure 3(a)). In our case, the (x, y) is the sky coordinate, R.A., and decl. When a group of detections belong to the same straight line, they would share common/similar (θ, r) (see Figure 3(b)). Therefore, we can explore the parameter space of (θ, r) for all the intranight detections on the master frame (see Figure 3(c)) and search for those sharing common (θ, r) to locate MBAs (see Figure 3(d).

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3.2. The Implementation

3.2.1. Detection Clean-up and Master Frame Generation

Before stacking the intranight detections of a field to create the mater frame, any detection was first removed from a source catalog if it is on the noisy area of the PS1 detector chip (Waters et al. 2016, see Figure 4 therein) or around any stationary source within 2''. Then, the remaining intranight detections were stacked into a master frame. According to the bulk motion of MBAs within 10 minutes, one asteroid detection was expected to have at least another companion on the master frame. Therefore, any detection on the master frame was further removed if it does not have any other neighbor within 15–30''. Typically, each field has ∼260,000 detections each night left on the master frame.

3.2.2. Hough Transform Calculation and Segment Identification on the Master Frame

After the clean-up step, the Hough transform, Equation (1), was applied to each detection on the master frame, in which r was calculated by exploring θ in a range of 1°–179° with a step of 01. Next, we searched for the detections sharing common (θ, r) to identify the line-up detections (hereafter, segment). Here, a common (θ, r) means the same value of θ and the same value of r until the fourth digit after the decimal point (i.e., within a 00001 difference).¹⁰ Only the segments with five or more detections were passed to the following processes. Typically, it takes ∼25 minutes to complete this step for a master frame on a computer cluster of five nodes.¹¹

3.2.3. Segment Combination on the Master Frame

In some cases, the neighboring segments on the master frame share one or more identical detections. When the difference in θ between these segments is less than 01, they are combined into a new segment, otherwise only the longest segment remained. Moreover, two segments on the master frame are combined if they have the following: (a) their mean locations can be mutually predicted within 7'', (b) their difference in θ is less than 01, and (c) their time spans have no overlap. A detection is further excluded from a segment when it has an incorrect epoch in the time sequence (i.e., simple increase/decrease in observation epochs) or has a magnitude 3σ away from the mean magnitude.

3.2.4. Segment Linkage across the Master Frames

To connect segments across different nights and fields, a linkage was performed to all the master frames. If two segments can mutually predict each other's mean locations within 60'' and have a difference in θ less than 01, they are identified as the same asteroid. In the end, the light curves of the remaining segments were extracted for further analysis. The flow chart of the Hough transform implementations is given in Figure 4.

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3.3. Orbital Determination and Diameter Estimation

To determine the preliminary orbits and absolute magnitudes in the w_P1 band of the newly discovered asteroids, Find_Orb was adopted.¹² Figure 5 shows the orbital distributions of the new asteroids along with the known ones in the observing fields. We see that the distributions are similar in the semimajor axis and inclination and, however, the new asteroids have a mean eccentricity slightly smaller than the known ones. This is because we only have four straight night arcs at the opposition, which provide relatively better constraints on the semimajor axis and inclination.¹³

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To estimate the diameters of the new asteroids, we used

$$\begin{eqnarray}&&D=\displaystyle \frac{1329}{\sqrt{{p}_{v}}}{10}^{-{H}_{w}/5},\end{eqnarray} \tag{ 2 }$$

where H_w is absolute magnitude in the w_p1 band, D is diameter in kilometers, p_v is geometric albedo in the V band, and 1329 is the conversion constant. We assumed p_v = 0.20, 0.08, and 0.04 for asteroids in the inner (2.1 < a < 2.5 au), mid (2.5 < a < 2.8 au), and outer (a > 2.8 au) main belts, respectively (Tedesco et al. 2005).¹⁴

To check the recovery rate of the Hough transform we used the known asteroids as of 2018 January 6 to compare with our result. The ephemerides of the known asteroids in the observing fields were obtained from the Jet Propulsion Laboratory/HORIZONS system, and then a cross-match was performed against the PS1 source catalogs to extract their detections using a radius of 2'' (hereafter, ephemeris-matching method). If an asteroid has five or more detections, we defined it as a successful recovery. In our observing fields there were 3870 known asteroids recovered by the ephemeris-matching method, out of which 3819 were also recovered by the Hough transform. For the asteroids lost by the Hough transform, we found that the detections of these objects that remained on the master frames are relatively small and, therefore, they have a lower chance to be recovered by the Hough transform. Two reasons account for this: (a) the detection number is intrinsically small (i.e., ∼80% of them have <20 detections using the ephemeris-matching method), and (b) the majority of the detections have been removed from the master frames in the clean-up step (i.e., ∼20% of them have tens of detections matched using the ephemeris-matching method).

We then compare the detection number for the known asteroids recovered by both methods. Figure 6 shows the distribution of the ratio of the detection number recovered by the Hough transform to that extracted by the ephemeris-matching method, where we see that the Hough transform has a smaller detection number on average. This is because (a) a relatively strict criterion has been applied in the definition of a segment in the Hough transform (i.e., a 00001 difference in r for a segment which is much smaller than the 2'' radius used in the ephemeris-matching method; see Section 3.2.2), and (b) some detections have been removed from the master frame in the clean-up step (see Section 3.2.1). However, a small portion of the known asteroids has a much smaller ratio of detection numbers (i.e., <50%). We found that these low-ratio asteroids have multiple segments in the Hough transform, which were not probably connected in the linking steps to become long segments (see the orange in Figure 6).

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4.1. Asteroid Discovery

In total, we found 3574 new asteroids that have five or more detections. Figure 7 shows the distribution of the detection number for these new asteroids, in which the 5–10 bin has many more objects. We believe that the high rise in that bin is mainly because some of these short segments were probably not connected in the linkage steps to be identified as the same asteroids, a situation similar to the multiple-segment known asteroids mentioned in Section 4. Despite these short-segment asteroids, the high detection number of the new asteroids suggests that they are unlikely to be false discoveries. Although it is probably difficult to trace them back at this moment due to the orbital uncertainty, we still submitted their detections to the Minor Planet Center in any case of accidental recovery in the near future. The summary table and the entire data set of light curve of these new asteroids can be found in Tables 2 and 3. Figure 8 shows the magnitude distribution of the new asteroids along with the known ones in our observing fields, where the new asteroids are mainly of 21.5 ≤ w_p1 ≤ 22.5 mag, ∼1 mag fainter than the samples of Chang et al. (2019; i.e., roughly 30% smaller in diameter). This suggests that the Hough transform has pushed the asteroid discovery to the actual limiting magnitude (∼22.5 mag) of the observation. With this magnitude range, we are able to reach 100 m MBAs (see Figure 9) and have a unique chance to conduct a rotation period study for them.

Table 2.  Information of the New Asteroids

ID	a (au)	e	i (◦)	Hw (mag)	D (km)
110000	2.34 ± 0.53	0.23 ± 0.20	1.08 ± 0.44	19.40	0.39
110006	2.33 ± 0.01	0.18 ± 0.00	1.92 ± 0.03	20.90	0.20
110007	3.11 ± 0.01	0.12 ± 0.01	9.73 ± 0.34	18.60	1.27
110012	2.78 ± 0.19	0.10 ± 0.11	6.90 ± 4.30	19.10	0.71
110015	2.54 ± 0.00	0.04 ± 0.01	3.53 ± 0.10	19.60	0.56
110020	2.58 ± 0.02	0.26 ± 0.00	4.72 ± 0.09	21.10	0.28
110022	2.66 ± 0.01	0.09 ± 0.01	2.60 ± 0.08	19.70	0.54
110023	2.32 ± 0.89	0.39 ± 0.25	5.62 ± 38.00	20.90	0.20
110025	2.39 ± 0.02	0.19 ± 0.02	0.96 ± 0.04	19.40	0.39
110029	2.55 ± 0.01	0.20 ± 0.02	3.90 ± 0.13	19.00	0.74
110037	3.13 ± 0.02	0.02 ± 0.02	3.80 ± 0.18	17.80	1.83
110038	2.37 ± 0.00	0.19 ± 0.02	5.76 ± 0.24	18.70	0.54
110045	2.51 ± 0.61	0.03 ± 0.21	4.05 ± 3.50	19.60	0.56
110046	2.76 ± 0.14	0.06 ± 0.13	2.34 ± 1.50	19.00	0.74
110051	2.43 ± 0.01	0.12 ± 0.01	1.84 ± 0.08	18.90	0.49
110063	3.04 ± 0.19	0.11 ± 0.13	1.49 ± 1.20	19.20	0.96
110066	3.07 ± 0.02	0.25 ± 0.01	5.85 ± 0.14	19.80	0.73
110069	3.14 ± 0.01	0.16 ± 0.03	10.22 ± 0.70	18.00	1.67
110071	2.54 ± 0.00	0.05 ± 0.01	0.77 ± 0.01	19.50	0.59
110077	2.88 ± 1.15	0.05 ± 0.27	1.41 ± 70.00	18.40	1.39

Note. Columns: temporary ID, semimajor axis (a, au), eccentricity (e), inclination (i, degree), absolute magnitude in the w_P1 band (mag), and calculated diameter (km).

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Table 3.  Light Curve Data Set

ID	MJD	R.A. (hms)	Decl. (dms)	m (mag)	Filter
110000	57687.29448	01 50 20.990	+12 38 28.43	21.92 ± 0.12	g
110000	57687.30355	01 50 20.415	+12 38 25.07	21.25 ± 0.05	w
110000	57687.31329	01 50 19.806	+12 38 21.58	21.51 ± 0.08	r
110000	57687.32231	01 50 19.221	+12 38 18.20	21.18 ± 0.05	w
110000	57687.33210	01 50 18.598	+12 38 14.66	20.92 ± 0.06	i
110000	57687.34120	01 50 18.015	+12 38 11.27	20.83 ± 0.04	w
110000	57687.35409	01 50 17.167	+12 38 06.42	20.63 ± 0.05	z
110000	57687.37634	01 50 15.737	+12 37 58.33	21.28 ± 0.06	g
110000	57687.38548	01 50 15.179	+12 37 55.01	20.91 ± 0.04	w
110000	57687.39549	01 50 14.536	+12 37 51.30	21.07 ± 0.05	r
110000	57687.46472	01 50 10.089	+12 37 25.56	20.80 ± 0.04	w
110000	57687.47452	01 50 09.459	+12 37 21.86	20.67 ± 0.05	i
110000	57688.28726	01 49 20.322	+12 32 20.68	21.03 ± 0.05	w
110000	57688.29380	01 49 19.916	+12 32 18.27	21.15 ± 0.06	w
110000	57688.30033	01 49 19.503	+12 32 15.93	21.14 ± 0.05	w
110000	57688.30687	01 49 19.090	+12 32 13.63	21.24 ± 0.05	w
110000	57688.31342	01 49 18.682	+12 32 11.20	21.33 ± 0.06	w
110000	57688.31996	01 49 18.256	+12 32 08.73	21.46 ± 0.07	w
110000	57688.32652	01 49 17.859	+12 32 06.31	21.53 ± 0.08	w
110000	57688.33311	01 49 17.414	+12 32 04.14	21.42 ± 0.07	w
110000	57688.33998	01 49 16.996	+12 32 01.26	21.40 ± 0.07	w
110000	57688.35428	01 49 16.085	+12 31 56.36	21.07 ± 0.04	w
110000	57688.36090	01 49 15.672	+12 31 54.00	21.00 ± 0.05	w
110000	57688.36753	01 49 15.246	+12 31 51.41	20.91 ± 0.05	w

Note. Columns: temporary ID, observation epoch (mjd), R.A. (hms), decl. (dms), magnitude, and PS1 filter.

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In general, the Hough transform is useful to discover asteroids when applied to the high-cadence observation. Moreover, its implementation is relatively easy and it works effectively on a data set containing massive noises.

Out of the 3574 new asteroids, 2853 objects, which have 10 or more detections in the w_P1 band in their light curves, were performed in a rotation period analysis using a second-order Fourier series fitting (Harris et al. 1989),

$$\begin{eqnarray}\begin{array}{rcl}{M}_{i,j} & = & \displaystyle \sum _{k=1}^{2}{B}_{k}\sin \left[\displaystyle \frac{2\pi k}{P}({t}_{j}-{t}_{0})\right]\\ & & +\,{C}_{k}\cos \left[\displaystyle \frac{2\pi k}{P}({t}_{j}-{t}_{0})\right]+{Z}_{i},\end{array}\end{eqnarray} \tag{ 3 }$$

where M_i,j are the reduced magnitudes in the w_P1 band measured at the epoch, t_j; B_k and C_k are the coefficients in the Fourier series; P is the rotation period; and t₀ is an arbitrary epoch. We also introduced a constant value, Z_i, to correct the possible offsets in magnitude between the measurements obtained from different nights.

In the end, we obtained 122 reliable rotation periods. Their information are summarized in Table 4 and their folded light curves can be found in Figures 10 and 11. Among the 122 objects, 22 have rotation periods of <2 hr, of which 13 objects were assigned as SFRs due to their highly convincing folded light curves (see Figure 12) and the other 9 objects were temporarily seen as SFR candidates (see Figure 13). The information of the SFRs and the SFR candidates are listed in Tables 5 and 6, respectively.

Table 4.  The 122 Reliable Rotation Periods

ID	a (au)	e	i (°)	Hw (mag)	D (km)	Period (hr)	$\bigtriangleup m$ (mag)	mw (mag)
110000	2.34 ± 0.53	0.23 ± 0.20	1.08 ± 0.44	19.40	0.39	2.71 ± 0.02*	0.48	21.19
110051	2.43 ± 0.01	0.12 ± 0.01	1.84 ± 0.08	18.90	0.49	5.48 ± 0.04	0.70	21.74
110066	3.07 ± 0.02	0.25 ± 0.01	5.85 ± 0.14	19.80	0.73	6.54 ± 0.07	0.57	21.97
110095	2.84 ± 0.01	0.11 ± 0.01	1.07 ± 0.01	18.60	1.27	4.08 ± 0.02	0.51	21.49
110139	2.30 ± 0.01	0.14 ± 0.01	22.79 ± 0.52	19.60	0.36	9.34 ± 0.07	0.67	21.81
110161	2.67 ± 0.01	0.10 ± 0.01	4.43 ± 0.20	18.50	0.94	6.50 ± 0.05	0.72	21.89
110174	2.74 ± 0.14	0.09 ± 0.13	4.98 ± 2.40	18.50	0.94	4.25 ± 0.03	0.61	21.86
110185	2.62 ± 0.12	0.13 ± 0.12	15.83 ± 6.00	18.50	0.94	5.12 ± 0.04	0.54	21.96
110208	2.41 ± 0.00	0.14 ± 0.02	1.04 ± 0.02	18.80	0.52	5.21 ± 0.02	0.84	21.83
110339	2.39 ± 0.01	0.18 ± 0.01	2.50 ± 0.04	20.20	0.27	4.96 ± 0.02	0.95	21.65
110503	2.98 ± 0.18	0.10 ± 0.14	0.59 ± 0.50	18.60	1.27	6.70 ± 0.04	0.77	21.62
110557	2.27 ± 0.01	0.15 ± 0.01	5.28 ± 0.09	20.10	0.28	5.50 ± 0.05	0.59	21.98
210158	3.19 ± 0.01	0.02 ± 0.01	11.33 ± 0.43	17.70	1.92	10.17 ± 0.18	0.50	21.80
210182	2.98 ± 0.00	0.04 ± 0.02	9.52 ± 0.33	18.50	1.33	5.00 ± 0.03	0.71	21.91
210315	2.67 ± 0.01	0.10 ± 0.01	2.24 ± 0.05	18.90	0.78	2.11 ± 0.01	0.61	21.70
210411	3.15 ± 0.02	0.06 ± 0.03	8.20 ± 0.31	18.20	1.52	15.38 ± 0.30	0.71	21.97
210566	3.14 ± 0.00	0.16 ± 0.01	1.29 ± 0.02	18.80	1.15	15.38 ± 0.30	0.64	21.68
210596	3.15 ± 0.03	0.09 ± 0.02	1.25 ± 0.03	17.60	2.01	6.63 ± 0.06	0.84	21.91
210776	2.43 ± 0.01	0.14 ± 0.01	2.10 ± 0.04	18.80	0.52	4.55 ± 0.02	0.74	21.64
220289	2.67 ± 0.02	0.30 ± 0.00	13.96 ± 0.15	20.80	0.33	3.71 ± 0.01	0.66	21.72
220355	3.17 ± 0.39	0.02 ± 0.16	15.97 ± 6.00	18.10	1.59	1.18 ± 0.00	0.49	21.95
220841	2.34 ± 0.18	0.21 ± 0.12	1.14 ± 0.37	20.00	0.30	1.53 ± 0.00	0.75	21.78
310006	2.58 ± 0.01	0.10 ± 0.02	9.50 ± 0.48	18.70	0.86	8.86 ± 0.13	0.66	21.81
310034	3.18 ± 0.01	0.14 ± 0.01	9.32 ± 0.32	17.60	2.01	5.63 ± 0.03	0.67	21.34
310149	2.48 ± 0.00	0.05 ± 0.01	2.68 ± 0.06	18.90	0.49	6.65 ± 0.04	0.61	21.58
310352	2.75 ± 0.02	0.23 ± 0.01	7.14 ± 0.18	19.40	0.62	1.32 ± 0.00	0.38	21.69
310361	2.72 ± 0.00	0.09 ± 0.02	8.85 ± 0.32	18.50	0.94	0.93 ± 0.00	0.41	21.92
310487	2.59 ± 0.00	0.09 ± 0.01	1.32 ± 0.02	18.90	0.78	6.72 ± 0.04	0.80	21.72
310509	2.52 ± 0.01	0.11 ± 0.01	14.09 ± 0.28	19.90	0.49	5.41 ± 0.04	0.60	21.94
310585	3.17 ± 0.03	0.10 ± 0.02	2.16 ± 0.11	17.80	1.83	6.00 ± 0.03	0.75	21.70
310657	2.18 ± 0.00	0.09 ± 0.01	6.26 ± 0.14	19.70	0.34	6.11 ± 0.05	0.88	21.84
310697	3.06 ± 0.01	0.10 ± 0.01	9.12 ± 0.34	18.20	1.52	6.61 ± 0.07	0.66	21.82
310754	5.70 ± 0.62	0.72 ± 0.04	2.10 ± 0.14	17.60	2.01	5.52 ± 0.03	1.04	21.57
310819	3.11 ± 0.02	0.09 ± 0.02	1.39 ± 0.06	18.10	1.59	5.30 ± 0.04	0.69	21.97
320362	3.10 ± 0.14	0.02 ± 0.10	16.36 ± 6.00	18.40	1.39	11.11 ± 0.16	1.35	22.01
410030	2.70 ± 0.01	0.10 ± 0.01	3.07 ± 0.08	19.40	0.62	4.61 ± 0.03	0.54	21.91
410071	2.47 ± 0.02	0.14 ± 0.02	2.43 ± 0.11	19.80	0.33	2.26 ± 0.01	0.38	21.58
410081	2.22 ± 0.03	0.21 ± 0.01	6.50 ± 0.19	21.00	0.19	1.66 ± 0.01	0.26	21.49
410110	2.45 ± 0.01	0.11 ± 0.01	9.94 ± 0.20	19.80	0.33	1.00 ± 0.00	0.72	21.82
410133	2.37 ± 0.00	0.15 ± 0.02	3.60 ± 0.15	19.10	0.45	6.56 ± 0.09	0.43	22.01
410139	2.23 ± 0.00	0.15 ± 0.01	2.46 ± 0.07	19.30	0.41	9.06 ± 0.10	0.66	21.83
410206	3.13 ± 0.02	0.23 ± 0.01	4.13 ± 0.10	19.20	0.96	5.70 ± 0.03	0.72	21.76
410295	2.65 ± 0.01	0.21 ± 0.01	5.73 ± 0.20	19.20	0.68	1.32 ± 0.00	0.45	22.09
410343	2.27 ± 0.01	0.16 ± 0.00	5.35 ± 0.07	21.00	0.19	13.56 ± 0.23	0.75	21.93
410761	2.85 ± 0.13	0.05 ± 0.13	6.21 ± 4.00	18.50	1.33	3.68 ± 0.02	0.46	21.80
410914	2.43 ± 0.01	0.18 ± 0.00	1.15 ± 0.01	20.40	0.25	3.99 ± 0.04	1.06	21.70
420204	2.80 ± 0.22	0.05 ± 0.14	6.66 ± 3.90	18.90	0.78	3.15 ± 0.03	0.47	22.12
420826	2.33 ± 0.01	0.19 ± 0.03	2.11 ± 0.08	19.30	0.41	1.38 ± 0.01	0.41	22.21
420849	2.59 ± 0.12	0.07 ± 0.10	11.87 ± 5.80	19.10	0.71	7.29 ± 0.04	1.02	21.73
510064	2.61 ± 0.01	0.07 ± 0.01	8.09 ± 0.23	19.20	0.68	7.16 ± 0.06	0.82	21.61
510160	3.06 ± 0.05	0.17 ± 0.06	9.26 ± 1.10	17.50	2.10	10.71 ± 0.19	0.50	21.73
510173	2.73 ± 0.01	0.11 ± 0.01	6.66 ± 0.24	19.10	0.71	7.36 ± 0.09	0.75	21.99
510201	2.32 ± 0.00	0.14 ± 0.01	6.62 ± 0.16	19.60	0.36	1.64 ± 0.01	0.45	22.02
510216	2.86 ± 0.00	0.07 ± 0.01	2.13 ± 0.05	18.60	1.27	3.79 ± 0.02	0.65	21.71
510235	2.42 ± 0.01	0.17 ± 0.01	1.83 ± 0.04	20.40	0.25	6.30 ± 0.07	0.69	22.01
510335	2.87 ± 0.01	0.01 ± 0.02	1.39 ± 0.02	18.40	1.39	5.53 ± 0.01	0.91	21.83
510350	2.67 ± 0.01	0.21 ± 0.01	1.45 ± 0.01	19.60	0.56	5.48 ± 0.05	0.43	21.71
510352	2.42 ± 0.01	0.19 ± 0.01	8.38 ± 0.25	20.10	0.28	1.00 ± 0.00	0.47	22.15
510391	2.91 ± 0.02	0.28 ± 0.01	16.19 ± 0.51	18.30	1.45	8.96 ± 0.14	0.77	21.59
510525	3.00 ± 0.14	0.14 ± 0.14	9.27 ± 4.60	17.90	1.75	0.61 ± 0.00	0.43	22.13
510545	2.86 ± 0.15	0.10 ± 0.13	1.47 ± 0.70	19.10	1.01	4.57 ± 0.04	0.60	21.99
510587	3.19 ± 0.01	0.09 ± 0.01	11.67 ± 0.36	18.20	1.52	3.10 ± 0.01	0.69	21.80
510629	2.86 ± 0.01	0.03 ± 0.01	1.83 ± 0.06	18.40	1.39	4.81 ± 0.04	0.68	21.92
510648	3.19 ± 0.02	0.28 ± 0.00	14.67 ± 0.28	19.60	0.80	31.58 ± 1.75*	1.21	21.95
510686	2.84 ± 0.15	0.05 ± 0.13	1.35 ± 0.70	18.80	1.15	7.41 ± 0.16	0.41	22.20
510737	2.33 ± 0.01	0.18 ± 0.00	1.13 ± 0.01	20.60	0.23	5.47 ± 0.03	0.61	21.54
510745	2.20 ± 0.00	0.11 ± 0.03	2.52 ± 0.09	18.80	0.52	0.55 ± 0.00	0.37	21.27
510746	3.13 ± 0.27	0.20 ± 0.16	2.31 ± 0.90	17.80	1.83	8.25 ± 0.11	0.54	21.98
520449	2.32 ± 0.01	0.13 ± 0.01	11.40 ± 0.28	20.00	0.30	0.63 ± 0.00	0.65	21.79
520489	2.75 ± 0.19	0.04 ± 0.13	6.47 ± 3.90	18.30	1.03	12.63 ± 0.27	0.55	21.61
520798	2.48 ± 0.01	0.20 ± 0.00	0.57 ± 0.01	20.40	0.25	0.77 ± 0.00	0.62	21.51
520810	2.40 ± 0.19	0.13 ± 0.15	3.11 ± 1.50	19.30	0.41	0.95 ± 0.01	0.35	22.27
610024	3.22 ± 0.02	0.17 ± 0.02	9.44 ± 0.44	18.20	1.52	6.15 ± 0.13	0.57	22.15
610183	3.06 ± 0.20	0.13 ± 0.15	0.80 ± 0.48	18.60	1.27	8.60 ± 0.09	0.98	21.67
610247	2.35 ± 0.00	0.07 ± 0.01	6.48 ± 0.15	19.50	0.37	13.41 ± 0.47	0.47	21.88
610248	2.72 ± 0.00	0.07 ± 0.01	0.75 ± 0.01	19.10	0.71	9.20 ± 0.11	0.65	21.82
610407	2.73 ± 0.01	0.10 ± 0.02	9.67 ± 0.43	18.50	0.94	2.59 ± 0.01	0.40	22.11
610554	2.84 ± 0.00	0.06 ± 0.02	5.94 ± 0.36	19.10	1.01	9.38 ± 0.15	0.58	22.02
610595	2.66 ± 0.01	0.11 ± 0.01	4.46 ± 0.13	18.50	0.94	2.52 ± 0.01	0.58	21.85
610679	2.71 ± 0.01	0.20 ± 0.01	13.79 ± 0.60	19.00	0.74	15.38 ± 0.40*	0.74	21.86
610700	3.17 ± 0.00	0.06 ± 0.02	10.14 ± 0.40	18.40	1.39	6.59 ± 0.05	0.73	21.87
610713	2.86 ± 0.00	0.05 ± 0.01	2.97 ± 0.11	19.00	1.05	6.74 ± 0.06	0.74	22.03
610731	2.91 ± 0.01	0.06 ± 0.01	1.58 ± 0.06	18.50	1.33	1.25 ± 0.01	0.40	22.08
610741	2.46 ± 0.01	0.19 ± 0.00	2.31 ± 0.03	20.70	0.22	7.29 ± 0.04	0.76	21.87
620515	2.20 ± 0.00	0.09 ± 0.01	3.64 ± 0.08	19.70	0.34	2.61 ± 0.01	0.48	22.04
620544	2.98 ± 0.21	0.03 ± 0.15	2.06 ± 1.10	18.40	1.39	2.87 ± 0.02	0.39	21.91
630389	2.34 ± 0.30	0.14 ± 0.16	2.78 ± 0.90	18.90	0.49	5.10 ± 0.07*	0.68	21.74
710033	3.02 ± 0.02	0.25 ± 0.01	7.59 ± 0.28	18.40	1.39	0.72 ± 0.00	0.56	22.18
710046	2.86 ± 0.01	0.06 ± 0.02	1.93 ± 0.06	18.40	1.39	4.67 ± 0.04	0.62	21.63
710060	2.85 ± 0.01	0.03 ± 0.01	1.77 ± 0.07	18.30	1.45	3.53 ± 0.02	0.48	21.67
710166	2.92 ± 0.01	0.05 ± 0.01	1.53 ± 0.03	18.30	1.45	7.32 ± 0.04	0.85	21.75
710172	2.42 ± 0.01	0.16 ± 0.00	0.69 ± 0.01	20.30	0.26	8.57 ± 0.12	0.67	21.78
710198	3.24 ± 0.03	0.29 ± 0.01	26.82 ± 0.80	18.40	1.39	6.03 ± 0.03	0.55	21.90
710342	2.29 ± 0.01	0.15 ± 0.00	0.80 ± 0.01	20.60	0.23	2.17 ± 0.01	0.49	21.70
710358	2.33 ± 0.01	0.21 ± 0.00	8.94 ± 0.09	21.00	0.19	5.45 ± 0.04	0.83	21.81
710385	2.78 ± 0.01	0.20 ± 0.01	1.90 ± 0.04	19.30	0.65	5.99 ± 0.05	0.64	22.21
710466	2.87 ± 0.00	0.05 ± 0.01	1.02 ± 0.02	18.80	1.15	7.29 ± 0.09	0.48	22.14
710523	2.89 ± 0.01	0.06 ± 0.01	1.64 ± 0.06	18.60	1.27	4.20 ± 0.03	0.54	22.10
710530	3.17 ± 0.03	0.16 ± 0.04	5.19 ± 0.24	17.20	2.41	9.27 ± 0.11	0.69	21.94
710624	2.80 ± 0.15	0.02 ± 0.14	6.13 ± 3.70	18.20	1.08	4.29 ± 0.02	0.69	21.45
710635	2.36 ± 0.01	0.20 ± 0.00	1.49 ± 0.01	20.70	0.22	5.41 ± 0.02	0.87	21.73
710649	2.35 ± 0.01	0.11 ± 0.01	4.39 ± 0.11	20.00	0.30	2.27 ± 0.01	0.47	22.04
710654	3.10 ± 0.03	0.16 ± 0.01	0.46 ± 0.01	18.20	1.52	1.05 ± 0.00	0.43	22.06
710661	3.16 ± 0.01	0.05 ± 0.03	9.97 ± 0.60	18.40	1.39	2.66 ± 0.04	0.40	22.09
720288	2.22 ± 0.01	0.04 ± 0.01	7.76 ± 0.20	20.00	0.30	13.11 ± 0.22*	0.72	21.92
720721	2.39 ± 0.02	0.22 ± 0.02	2.84 ± 0.10	20.00	0.30	1.46 ± 0.01	0.43	22.17
810046	2.68 ± 0.01	0.13 ± 0.01	3.60 ± 0.10	18.90	0.78	5.47 ± 0.05	0.36	21.67
810074	2.76 ± 0.01	0.17 ± 0.01	8.84 ± 0.26	18.40	0.98	14.04 ± 0.25	0.62	21.83
810085	3.08 ± 0.01	0.03 ± 0.01	10.43 ± 0.34	18.00	1.67	15.58 ± 0.21	0.84	21.77
810092	2.76 ± 0.04	0.31 ± 0.00	12.51 ± 0.27	21.00	0.30	4.36 ± 0.04	0.64	22.04
810154	3.96 ± 0.07	0.06 ± 0.04	12.89 ± 0.90	17.00	2.65	8.89 ± 0.17	0.53	22.00
810213	3.02 ± 0.02	0.27 ± 0.01	2.48 ± 0.06	19.60	0.80	8.66 ± 0.13	0.57	21.75
810316	2.61 ± 0.03	0.28 ± 0.02	6.32 ± 0.09	21.00	0.30	1.67 ± 0.01	0.63	22.03
810398	2.93 ± 0.17	0.02 ± 0.15	3.13 ± 2.00	18.30	1.45	5.10 ± 0.03	0.55	21.75
810464	2.19 ± 0.01	0.10 ± 0.01	7.91 ± 0.15	20.60	0.23	4.43 ± 0.05	0.53	21.96
810534	2.16 ± 0.01	0.07 ± 0.00	4.39 ± 0.07	20.30	0.26	5.96 ± 0.04	0.52	21.67
810554	2.60 ± 0.01	0.17 ± 0.01	2.53 ± 0.05	19.90	0.49	5.76 ± 0.06	0.84	21.93
810586	2.89 ± 0.02	0.19 ± 0.01	13.45 ± 0.34	19.80	0.73	0.71 ± 0.00	0.53	22.18
810747	2.74 ± 0.14	0.03 ± 0.14	6.25 ± 4.00	18.10	1.13	5.91 ± 0.04*	0.89	21.38
810752	2.61 ± 0.02	0.30 ± 0.01	2.88 ± 0.06	18.40	0.98	3.64 ± 0.02	0.44	21.88
810762	2.91 ± 0.00	0.06 ± 0.01	2.11 ± 0.05	18.60	1.27	4.27 ± 0.02	0.56	21.94
830672	2.29 ± 0.31	0.17 ± 0.17	3.60 ± 2.70	21.00	0.19	0.51 ± 0.00	0.39	22.25

Note. Columns: temporary ID, semimajor axis (a, au), eccentricity (e, degree), inclination (i, degree), absolute magnitude in the w_p1 band (H, mag), diameter (D, km), derived rotation period (hr), light curve amplitude (mag), and mean apparent magnitude in the w_p1 band (mag). Note that half periods are marked with "*."

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Table 5.  The 13 Super-fast Rotators

ID	a (au)	e	i (◦)	Hw (mag)	D (km)	Period (hr)	$\bigtriangleup m$ (mag)	mw (mag)
220841	2.34 ± 0.18	0.21 ± 0.12	1.14 ± 0.37	20.00	0.30	1.53 ± 0.00	0.75	21.78	20
310352	2.75 ± 0.02	0.23 ± 0.01	7.14 ± 0.18	19.40	0.62	1.32 ± 0.00	0.38	21.69	80
310361	2.72 ± 0.00	0.09 ± 0.02	8.85 ± 0.32	18.50	0.94	0.93 ± 0.00	0.41	21.92	438
410110	2.45 ± 0.01	0.11 ± 0.01	9.94 ± 0.20	19.80	0.33	1.00 ± 0.00	0.72	21.82	64
510352	2.42 ± 0.01	0.19 ± 0.01	8.38 ± 0.25	20.10	0.28	1.00 ± 0.00	0.47	22.15	36
510525	3.00 ± 0.14	0.14 ± 0.14	9.27 ± 4.60	17.90	1.75	0.61 ± 0.00	0.43	22.13	3880
520449	2.32 ± 0.01	0.13 ± 0.01	11.40 ± 0.28	20.00	0.30	0.63 ± 0.00	0.65	21.79	132
520798	2.48 ± 0.01	0.20 ± 0.00	0.57 ± 0.01	20.40	0.25	0.77 ± 0.00	0.62	21.51	57
610731	2.91 ± 0.01	0.06 ± 0.01	1.58 ± 0.06	18.50	1.33	1.25 ± 0.01	0.40	22.08	438
710033	3.02 ± 0.02	0.25 ± 0.01	7.59 ± 0.28	18.40	1.39	0.72 ± 0.00	0.56	22.18	1995
710654	3.10 ± 0.03	0.16 ± 0.01	0.46 ± 0.01	18.20	1.52	1.05 ± 0.00	0.43	22.06	889
810316	2.61 ± 0.03	0.28 ± 0.02	6.32 ± 0.09	21.00	0.30	1.67 ± 0.01	0.63	22.03	13
810586	2.89 ± 0.02	0.19 ± 0.01	13.45 ± 0.34	19.80	0.73	0.71 ± 0.00	0.53	22.18	537

Note. Columns: temporary ID, semimajor axis (a, au), eccentricity (e, degree), inclination (i, degree), absolute magnitude in the w_p1 band (H, mag), diameter (D, km), derived rotation period (hr), light curve amplitude (mag), mean apparent magnitude in the w_p1 band (mag), and estimated cohesion (Pa).

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Figure 14 shows the plot of the diameters versus rotation periods for these 122 objects on top of the samples obtained from Chang et al. (2019) and the U ≥ 2 objects in the light curve database (LCDB; Warner et al. 2009). These SFRs are unlikely to be explained by the rubble-pile structure unless they have unusual high bulk density. Compared to the similar studies, e.g., Chang et al. (2015, 2016, 2019), the chance of finding SRFs is higher here. As shown in the simulation carried out by Chang et al. (2019, see Figure 12 therein), the recovery rates are similar for the spin rates of ≥3 rev day⁻¹ within each magnitude interval.¹⁵ This suggests that more SFRs found here are not because we are in favor of detecting short periods. Therefore, we believe that small MBAs probably harbor more SFRs. One possible explanation is that these SFRs are monoliths and, moreover, small asteroids are more likely to be monoliths. Consequently, more SFRs could be expected in small MBAs. However, it is difficult to explain why the relatively large SFRs (i.e., diameter of ∼1 km) could avoid numerous collisions after their formations and remain monolithic. Another possible explanation is the size-dependent cohesion model in which the cohesion plays a significant role for small asteroids, allowing them to rotate faster without breaking apart (Holsapple 2007). This is another way to expect more SFRs in small asteroids. We adopted the equations shown in Holsapple (2007), Rozitis et al. (2014), Polishook et al. (2016), and Chang et al. (2017) and assumed a bulk density of ρ = 2 g cm⁻³ to estimate the cohesion required to survive these SFRs under their super-fast spinning. The calculated cohesion of each SFR are given in Tables 5 and 6, in which most cases need a cohesion of tens to thousands of Pa, a value consistent with the previously reported SFRs (see Table 4 in Chang et al. 2019) and the regolith on the Moon and Mars (such as the Moon and Mars; Mitchell et al. 1974; Sullivan et al. 2011). However, two SFRs, 510525 and 710033, need a cohesion up to thousands of Pa, which is several times larger than the average value. This kind of SFR, larger than a kilometer and a rotation period of <1 hr, can put a critical constraint on the cohesion model. Therefore, it is encouraged to conduct a comprehensive rotation period survey on the asteroids of few kilometers in size.

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The intranight detections of asteroids show up as straight lines and, therefore, the Hough transform can be used to locate these line-up detections to discover new asteroids. We applied this algorithm to the PS1 high-cadence observation, a survey that was originally conducted to collect a large sample of asteroid light curves for rotation period measurement during 2016 October 26–31 (Chang et al. 2019). Most of the known asteroids in the observing fields were recovered and, moreover, 3574 new asteroids, mainly of 21.5 < m < 22.5 mag, were found. Using the light curves of the new asteroids, we obtained 122 reliable rotation periods, of which 13 are SFRs. Compared to the previous surveys, the chance of discovering SFRs is much higher here (i.e., ∼11%). Since the data set used here is the same as Chang et al. (2019), except our samples are mainly of 21.5 ≤ w_p1 ≤ 22.5 mag (i.e., one magnitude fainter than theirs; ∼30% smaller in size), our result suggests that subkilometer MBAs possibly harbor more SFRs.

This work is equally contributed by C.-K.C. and K.-J.L. This work is supported by the National Science Council of Taiwan under the grants MOST 107-2112-M-008-009-MY2 and MOST 108-2112-M-008-004. We thank the anonymous referee for the comments and suggestions, which greatly improved this work.