Introduction

From the asteroid spin-rate distribution, we can understand the mechanisms altering asteroid spin status. In addition, the asteroid spin-rate limit also provides a practical way to investigate the asteroid interior structure. However, a comprehensive understanding of the aforementioned questions depends on a large sample of asteroid spin rates. Thanks to advanced technology—wide-field detectors, high computing power, massive data storage, and robotic observation—it is possible to obtain numerous asteroid light curves within a short period of time to derive hundreds of asteroid rotation periods.

It is believed that asteroids with diameter of \(D > 0.15\) km have “rubble-pile” structure, in which asteroids are gravitationally bounded aggregations with no assumed internal cohesion. Therefore, a rubble-pile asteroid cannot rotate exceedingly fast, otherwise the centrifugal force would overcome the gravity, which consequently would lead to the breakup of the asteroid. This idea is supported by the “2.2-h spin barrier”Footnote 1 held for asteroids of \(D > 0.15\) km (Harris 1996; Pravec and Harris 2000). On the other hand, the asteroids of \(D < 0.15\) km can rotate faster than the 2.2-h spin barrier because of their monolithic structure. However, 2001 OE84, an asteroid of \(D \sim 0.65\) km that completes 1 rotation in 29.19 min, demonstrates the first exception among the rubble-pile asteroids (Pravec et al. 2002), and it is, therefore, called a “super-fast rotator” (SFR). To explain the presence of 2001 OE84, Holsapple et al. (2007) introduced the concept of internal cohesive strength into the rubble-pile structure. This idea allows a transition zone between monolithic and rubble-pile asteroids to accommodate SFRs. If this model is realistic, a certain number of SFRs should be found among the rubble-pile asteroids. Therefore, the physical properties of SFRs provide important information for understanding asteroid interior structure. Moreover, another important implication of rubble-pile structure is that the lower the bulk density of an asteroid, the longer its spin barrier can be expected.Footnote 2 With enough samples of rotation period and taxonomy type, we should be able to see different spin-rate limits for different types of asteroids.

In addition, we can study how the spin status of asteroids can be affected by mutual collisions and the Yarkovsky–O’Keefe–Radzievskii–Paddack (YORP) effect from the spin-rate distribution of asteroid. When an asteroid system stays in a collisional equilibrium state, it should have a Maxwellian spin-rate distribution (Salo 1987), which has been observed indeed for asteroids of \(D > 40\) km (Pravec et al. 2002). To the contrary, a number excess was found both in the slow and fast ends of the spin-rate distribution for asteroids of \(3< D < 15\) km (i.e., a flat form, the number in each spin-rate bin is similar) (Pravec et al. 2008), which indicates a non-collisional equilibrium system. It is believed that the YORP effect, which can alter the spin rates of asteroids of few kilometers in size within a million-year timescale (Rubincam 2000), creates the aforementioned excesses in the both ends of the distribution (Pravec et al. 2008). However, more recent results from large sky surveys show a different spin-rate distribution of asteroids of \(3< D < 15\) km, which is more like a Maxwellian form (Masiero et al. 2009; Polishook et al. 2012). These two forms of distributions of asteroids of \(3< D < 15\) km suggest different scenarios of spin-status evolution. Although the YORP effect accounts for both distributions (e.g., flat distribution vs. Maxwellian-like distribution), a different degree of spin-status alteration would be required for each case.

Along this story line, we initiated our asteroid spin-rate study using the iPTFFootnote 3 to collect numerous asteroid light curves with wide sky coverage since February 2013. As a report presented at the AOGS 2016 conference, this article summarizes our work conducted over the past 3 years. We organize this article in the following way: a brief introduction of the data survey and rotation-period analysis is given in "Data survey and rotation-period analysis" section; the results and discussion are presented in "Results and discussion" section; and the summary and conclusions can be found in "Summary" section.

Data survey and rotation-period analysis

Data survey

The iPTF is a project to explore the transient and variable sky synoptically. Utilizing the Palomar 48-inch Oschin Schmidt Telescope and an 11-chip mosaic CCD camera, the iPTF has a field of view of \({\sim}7.26\)  deg\(^2\) (Law et al. 2009; Rau et al. 2009). Most of the iPTF exposures were taken in the Mould-R band, and the other available filters include the Gunn-g’ and four different \(H_{\upalpha }\) bands. The exposure time of the iPTF is fixed to 60 s, which can routinely reach a limiting magnitude of \(R\, \sim 21\) mag at the \(5\sigma\) level (Law et al. 2010).

All iPTF exposures are processed by the IPAC-PTF photometric pipeline (Grillmair et al. 2010; Laher et al. 2014). The absolute magnitude is calibrated against Sloan Digital Sky Survey fields (SDSS) (York et al. 2000), and the accuracy can reach \({\sim} 0.02\) mag for the so-called photometric nights (Ofek et al. 2012a, b).

In order to collect a large number of asteroid light curves, we initiated a survey for asteroid spin rates using the iPTF since February 2013. Until early in 2015, we carried out eight campaigns, including (1) three sets of observations of 12 fields at 20-min cadence in four adjacent nights, in 15–18 February 2013, 6–9 January 2014 , and 20–23 February 2014 (Chang et al. 2014a, b, 2015); and (2) four sets of observations of 6 fields at 10-min cadence in two/three adjacent nights, in 29–31 October 2014, 10–13 November 2014 , 18–19 January 2015 , 20–21 February 2015, and 25–26 February 2015 (Chang et al. 2016). The total sky coverage was \({\sim} 450\) deg\(^2\). More details about each campaign are available in the original published papers, please see (Chang et al. 2014a, b, 2015, 2016).

To extract the light curves of known asteroids, we first removed all stationary sources from the iPTF catalogs, and then matched all residual detections against the ephemerides obtained from the JPL/HORIZONS system within a search radius of two arc seconds. We also excluded all detections flagged as defective by the IPAC-PTF photometric pipeline from the extracted light curves. We finally obtained \({\sim} 17{,}000\) asteroid light curves, each with number of detections \({\ge} 10\) (hereafter, PTF-detected asteroids), which were then subjected to period analysis (see below for details). The magnitude distribution of the PTF-detected asteroids is given in Fig. 1, which ranges from \({\sim} 14\) to \({\sim} 21\) mag; a comparable histogram is also shown for the smaller subset of asteroids that we determined to have reliable rotation periods (hereafter, PTF-U2s; see next subsection for more details).

Fig. 1
figure 1

This figure is reproduced from Chang et al. (2015, 2016)

Magnitude distribution of the PTF-detected asteroids. Gray the PTF-detected asteroids. Red the PTF-U2s.

Full size image

We adopted the diameter estimations of the WISE/NEOWISE measurements (Grav et al. 2011; Mainzer et al. 2011; Masiero et al. 2011), if available, for PTF-detected asteroids. Otherwise, the diameters were estimated using

$$\begin{aligned} D = {1130 \over \sqrt{p_R}} 10^{-H_R/5}, \end{aligned}$$
(1)

where D is the diameter in km, \(H_R\) is the R band absolute magnitude, \(p_R\) is the R band geometric albedo, and the conversion constant of 1130 is from Jewitt et al. (2013). We assumed three empirical albedo values in the estimation, which are \(p_R =\) 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).Footnote 4 Figure 2 shows a plot of semi-major axes vs. diameters for the PTF-U2s, where we see that the asteroid diameter range changes along with the semi-major axis due to our limiting magnitude. Therefore, we have only a few asteroids of \(D > 15\) km in the inner main belt. The same situation also occurs for the asteroids of \(D < 3\) km in the outer main belt.

Fig. 2
figure 2

This figure is reproduced from Chang et al. (2015, 2016)

Diameter vs. semi-major axis of the PTF-U2s. Most of the PTF-U2s are main-belt asteroids. Because of the limiting magnitude (i.e., \({\sim} 14\) to \({\sim} 21\) mag), the diameter range changes with semi-major axis. The albedo values adopted in the diameter calculations are indicated in the plot.

Full size image

Rotation-period analysis

To measure the asteroid rotation periods, all light-curve measurements were corrected for light-travel time and placed at both heliocentric, r, and geocentric, \(\triangle\), 1 AU distances to obtain reduced magnitudes. The phase angles underwent only a small change during our short observation time span for each survey (i.e., \({\le}4\) days). Therefore, we estimate the absolute magnitude simply by applying a fixed \(G_R\) slope of 0.15 in the H–G system (Bowell et al. 1989). Then, the traditional second-order Fourier series method was used to derive rotation periods (Harris et al. 1989):

$$\begin{aligned} M_{i,j} &= \sum _{k=1,2} \left\{ B_k\sin \left[ \frac{2\pi k}{P} (t_j-t_0)\right]\right. \\ & \quad\quad\left.+ \; C_k\cos \left[ \frac{2\pi k}{P} (t_j-t_0)\right] \right\} + Z_i, \end{aligned}$$
(2)

where \(M_{i,j}\) are the R band reduced magnitudes measured at the light-travel-time-corrected epoch, \(t_j\); \(B_k\) and \(C_k\) are the Fourier coefficients; P is the rotation period; \(t_0\) is an arbitrary epoch; and the constant values, \(Z_i\), are the zero-point offsets between measurements obtained from different fields, nights, and chips. Since the equation has at least seven to nine parameters to be fitted, we therefore confine our period analysis to the light curves with \({\ge} 10\) detections. All the derived rotation periods were reviewed by careful eye inspection to determine their reliability according to their folded light curves. We follow the definition in Warner et al. (2009) and assigned a quality code to each derived rotation period, where ‘3’ means highly reliable; ‘2’ means some ambiguity; ‘1’ means possible, but may be wrong. Moreover, when we were unable to find any acceptable solution for a light curve, it was assigned \(U = 0\). We finally harvested 2780 reliable rotation periods (i.e., \(U \ge 2\); hence, the reason we name them PTF-U2s) from our pool of \({\sim} 17,000\) PTF-detected asteroids.

Results and discussion

Spin-rate limit and super-fast rotators

From our surveys, we identified two new SFRs. Our first SFR is (335433) 2005 UW163, which is a V-type asteroid with a diameter of \({\sim} 0.6\) km that completes one rotation in 1.29 h (Chang et al. 2014, 2015). Its super-fast rotation was initially identified in the 2014 February survey using the iPTF and then confirmed by 2014 March follow-up observations using the 200-inch Hale Telescope at the Palomar Observatory. The identification of 2005 UW163 as the second SFR after 2001 OE84 ensures the existence of SFR group. Our second SFR is (40511) 1999 RE88, which is a S-type asteroid with a diameter of \({\sim} 2\) km that completes one rotation period in 1.96 h (Chang et al. 2016). In addition, we also found 23 SFR candidates. Figure 3 shows a plot of rotation periods vs. diameters for the PTF-U2s on top of the data points of \(U \ge 2\) obtained from the Asteroid Light Curve Database (LCDB)Footnote 5 (Warner et al. 2009).

Fig. 3
figure 3

This figure is reproduced from Chang et al. (2015, 2016)

Asteroid rotation period vs. diameter. The red and gray filled circles are the PTF-U2s and the LCDB objects of \(U \ge 2\), respectively. The larger blue filled symbols are the five known SFRs. The green filled circles are the 23 SFR candidates. The dashed line is the predicted spin barrier adopted from Holsapple et al. (2007), where cohesion is \(\kappa = k\bar{r}^{-1/2}\) and \(k = 2 \times 25 ^7\) dynes cm\(^{-3/2}\). Note that the uncertainty in diameter estimation using Eq. (1) is \({\sim} 20\%\), which includes the uncertainty in the absolute magnitude H. The uncertainty in the rotation period is smaller than the symbol size.

Full size image

In general, we see that the majority of rubble-pile asteroids (i.e., \(D > 0.15\) km) are all below the 2.2-h spin barrier. Some monolithic objects (i.e., \(D < 0.15\) km), however, can have rotation periods shorter than 2.2 h. In addition, the five SFRs, including 2001 OE84 (Pravec et al. 2002), 2000 GD65 (Polishook et al. 2016), and 1950 DA (Rozitis et al. 2014), and the 23 candidates stand out from the rubble-pile asteroids to form a group above the 2.2-h spin barrier that cannot be explained by the rubble-pile structure unless they have anomalously high bulk densities (see Fig.  4). To keep these five SFRs from breaking up, it requires an internal cohesion ranging from several tens to several hundreds of Pa (Chang et al. 2016). This value is similar to that of lunar regolith (i.e., 100 to 1000 Pa) (Mitchell et al. 1974). However, the cohesion for those candidates with relatively large diameters is up to several thousands of Pa, which is almost an order of magnitude larger than that of the five SFRs. Therefore, to elucidate the candidates with relatively large diameters is important to understand asteroid interior structure. When compared with the average asteroids, SFRs are very rare. If internal cohesion can be universally applied to asteroids, we should be able to identify more SFRs. This probably suggests that the SFRs is a peculiar group apart from the typical asteroids.

Fig. 4
figure 4

This figure is reproduced from Chang et al. (2015, 2016)

Light-curve amplitude vs. spin rate. The symbols are the same as in Fig. 3. The dashed, dot-dashed, and dotted lines represent the spin-rate limits, \(P \sim 3.3 \sqrt{(1 + \Delta m)/\rho }\) (Pravec and Harris 2000), for rubble-pile asteroids with a bulk densities of \(\rho =\) 3, 2, and 1 g cm−3, respectively. Note that the asteroids of \(D < 0.2\) km are not included in this plot.

Full size image

Spin rate vs. spectral type

In order to know whether the asteroids of different spectral types have various spin-rate limits, we select the asteroids from our PTF-U2s and the LCDB based on both rotation period and SDSS spectral type. We finally have 635 C-type asteroids and 1227 S-type asteroids.Footnote 6 The left column in Fig. 5 shows plots of light-curve amplitudes vs. spin rates for the C- and S-type asteroids (upper and lower panels, respectively). The C-type asteroids appear to have a boundary around the spin-rate limit for \(\rho = 1.5\) g cm\(^{-3}\) and only a few C-type asteroids can reach the spin-rate limit for \(\rho = 2\) g cm\(^{-3}\). On contrary, the S-type asteroids show a clear boundary close to the spin-rate limit for \(\rho = 2\) g cm\(^{-3}\). While the C-type has only 22 asteroids (i.e., \({\sim} 3 \%\)) over the spin-rate limit for \(\rho = 1.5\) g cm\(^{-3}\), the S-type has 147 (i.e., \({\sim} 12 \%\)) asteroids over that limit. This difference becomes more clear in the spin-rate distributions of the C- and S-type asteroids with \(3< D < 20\) km, which is shown in the right panel of Fig. 5. We see that the C-type asteroids have much fewer objects than the S-type for \(f > 8\) rev/day. Our results show that the C-type asteroids have a lower spin-rate limit than the S-type asteroids. This is in good agreement with the general picture of rubble-pile structure, in which asteroids with a lower bulk density have a lower spin-rate limit. A similar result was also obtained using the asteroid rotation periods derived from the iPTF archive data (Waszczak et al. 2015). We also suspect that those C-type asteroids with higher spin rates (i.e., \(f > 8\) rev/day) could be of the M- or E-type asteroids, which cannot be separated from C-type asteroids using the SDSS color. The M- or E-type asteroids have a higher bulk density, and, therefore, they have a higher spin-rate limit.

Fig. 5
figure 5

This figure is reproduced from Chang et al. (2015, 2016)

Light-curve amplitude vs. spin rate (left) and spin-rate distribution (right) for the C- (upper) and the S-type (lower) asteroids. The gray and red filled circles are objects of the LCDB and the PTF-U2s, respectively. The number of asteroids used in the statistics is given in each plot. The spin-rate limits, \(P \sim 3.3 \sqrt{(1 + \Delta m)/\rho }\) (Pravec and Harris 2000), are represented by dashed, dot-dashed, and dotted lines for rubble-pile asteroids with bulk densities of \(\rho =\) 3, 2, and \(1\,\text{g}\,\text{cm}^{-3}\), respectively. The spin-rate distributions use only asteroids of \(3< D < 20\) km.

Full size image

The spin-rate distributions

Using the asteroid rotation periods acquired from the iPTF surveys, we are now able to generate the spin-rate distribution as a function of size and location in the main belt. We also take into account the observational bias in our surveys, in which synthetic light curves are used to determine our ability to derive rotation periods according to the survey conditions (see Chang et al. 2015 and the references therein). Figures 6 and 7 show the spin-rate distributions from the 20- and 10-min cadence surveys, respectively. The distributions are separated out for asteroids with \(3< D < 15\) km and \(D < 3\) km at the inner (\(2.1< a < 2.5\) AU), middle (\(2.5< a < 2.8\) AU), and outer (\(a > 2.8\) AU) main belt. Since the time span of each campaign was only a few days (i.e., 2–4 days depending on the campaign), we were unable to recover relatively long rotation periods (i.e., spin rate \(f \le 3\) rev/day). Therefore, the numbers in the distribution bins for \(f \le 3\) rev/day are greatly biased.

Fig. 6
figure 6

This figure is reproduced from Chang et al. (2015)

Spin-rate distributions of the 20-min cadence surveys. The spin-rate distributions for asteroids of \(3< D < 15\) km (upper) and \(D < 3\) km (lower) at inner (left), middle (center), and outer (right) main belt. The solid black line and blue dashed line are the original and de-biased results, respectively. The number of asteroids used in each distribution is indicated in each plot. Note that the number of asteroids of \(D > 15\) km and that of \(D < 3\) km in outer main belt are too small to carry out meaningful statistics.

Full size image
Fig. 7
figure 7

This figure is reproduced from Chang et al. (2016)

Spin-rate distributions of the 10-min cadence surveys. The spin-rate distributions for asteroids with diameter of \(3< D < 15\) km (upper) and \(D < 3\) km (lower) at inner (left), middle (center), and outer (right) main belt. The solid black line and blue dashed line are the original and de-biased results, respectively. Note that the number of asteroids of \(D > 15\) km and that of \(D < 3\) km in outer main belt are too small to carry out meaningful statistics.

Full size image

For asteroids with \(3< D < 15\) km, we see that the spin-rate distribution decreases steadily with frequency for \(f > 5\) rev/day regardless of their locations in the main belt. Our result is different from the flat distribution shown by Pravec et al. (2008). This suggests that the timescale of the YORP effect to alter asteroid spin rates is probably longer than expected.

For asteroids with \(D < 3\) km, we note an abrupt number drop from \(f = 5\) rev/day to \(f = 6\) rev/day in the spin-rate distributions. We also find fewer large-amplitude objects (i.e., \(\Delta m > 0.2\) mag) in the bins of \(f > 6\) rev/day. Because the timescale of the YORP effect is smaller for asteroids of \(D < 3\) km, and, moreover, the spin-rate limit is lower for asteroids with large amplitude, we believe that these large-amplitude small asteroids might have been spun up to reach their lower spin-rate limit and, as a consequence, have been broken up into smaller, harder-to-detect pieces. This explains the aforementioned steep number drop, as well.

Summary

To look for SFRs and to conduct a comprehensive study on asteroid rotation, we initiated a project to survey asteroid rotation periods using the iPTF since February 2013. To date, eight campaigns have been carried out, and valuable data have been acquired.

From our survey, we identified two SFRs, (335433) 2005 UW163 and (40511) 1999 RE88, and 23 candidates. (335433) 2005 UW163 is the second SFR to be discovered, which ensures the existence of the SFR group. However, the population size of SFRs is very small compared to that of average asteroids.

The internal cohesion of those candidates with relatively large diameters (i.e., \(D > 3\) km) is up to thousands of Pa. This value is an order of magnitude larger than that for the five known SFRs. Future confirmation of their super-fast rotations will provide important information about asteroid interior structure.

With the collected asteroid rotation periods, we found a lower spin-rate limit for C-type asteroids (i.e., lower bulk density) when compared with S-type asteroids. This follows the general concept of rubble-pile asteroids in that the lower their bulk density, the lower their spin-rate limit.

We generated spin-rate distributions for asteroids according to their size and locations in the main belt. The spin-rate distributions of asteroids of \(3< D < 15\) km decrease steadily with frequency for \(f > 5\) rev/day, regardless of their locations in the main belt. This might suggest that the timescale of the YORP effect to alter asteroid spin rate is longer than expected. We also find a significant number drop from \(f=5\) to \(f=6\) rev/day in the spin-rate distributions for asteroids of \(D < 3\) km. Moreover, the results show a shortfall in the number of large-amplitude, small-size asteroids for \(f > 6\) rev/day. The possible explanation is that the large-amplitude, small-size asteroids have been spun up to reach their lower spin-rate limit, and, consequently, this led to them breaking apart.