The distribution of near-Earth asteroid (NEA) rotation rates differs considerably from the similar distribution of Main Belt asteroids (MBAs) by the presence of excesses of fast and slow rotators, which are not observed or not so prominent in the distribution for MBAs. Among possible reasons for the difference, there can be influence of solar radiation on spin rate of small NEAs, the so-called “YORP effect,” which appears due to reflection, absorption, and IR re-emission of the sunlight by an irregularly shaped rotating asteroid. It is known that the YORP-effect action strongly depends on the amount of solar energy obtained by the body (insolation), its size, and albedo. The analysis of observation data has shown that: (1) the mean diameter of NEAs decreases from the middle of the distribution to its ends, that is, the excesses of slow rotators (ω ≤ 2 rev day−1) and fast rotators (ω ≥ 8 rev day−1) are composed of smaller NEAs than in the middle of the distribution; (2) NEAs of both excesses are in the orbits where their insolation is about 8–10% larger than that of NEAs in the middle of the distribution; and (3) the objects in both excesses have a little lower albedo on average than that of objects in the middle of the distribution. All these results qualitatively agree well with the YORP-effect action and may be considered as independent arguments in favor of it.
It is well known that distribution of rotation rates for near-Earth asteroids (NEAs) differs essentially from the same distribution for Main Belt asteroids, especially for larger ones, because it shows clear excesses of fast and slow rotators (Pravec and Harris 2000; Harris and Pravec 2006; Lupishko et al. 2007; Pravec et al. 2008). Among the possible reasons for the difference, one can mention the difference in size distribution of bodies in these populations, influence of close encounters with inner planets and nongravitational (radiation) effects, influence of rotational parameters of binary systems, and possibly some other effects including the selection ones. The contribution of the selection effects can be essential because the population of small MBAs is much less complete than that of the small NEAs. In particular, Polishook and Brosch (2009) note that the spin rate distributions of NEAs and the smallest MBAs (D < 5 km) are very similar, although, as it will be shown below (Figs. 1a and 1b), the similarity is not complete. The fact that these excesses are at the high and low ends of the distribution suggests an idea that some stable mechanism (or mechanisms) exists, which results in acceleration or deceleration of NEA rotation. Some recent studies (see, e.g., Rossi et al. 2009 and references therein) are evidence that one of the most probable mechanisms could be so-called YORP effect (Rubincam 2000), which appears because of reflection and absorption of solar radiation and its IR re-emission by an irregular-shaped body. The term “YORP” is an abbreviation created from initial letters of the authors’ surnames (Yarkovsky–O'Keefe–Radzievskii–Paddack effect), who developed the general theory of Yarkovsky effect and applied it to rotation of small bodies.
It is known that the action of this effect depends on the size of the asteroid (the smaller the size of body, the stronger the influence of the effect on its rotation), its shape, and on the amount of solar radiation (insolation) that body receives (Rubincam 2000; Bottke et al. 2006). At the same time, an increment of angular momentum may be both positive and negative. There is no doubt that the action of the effect is very weak, but it is accumulated during the whole lifetime of NEA (approximately 107 yrs) and therefore can result in noticeable change in spin rate and obliquity of small NEAs. Currently, there are experimental verifications and magnitude estimations of the YORP-effect influence on the rotation of the four NEAs: 1820 Apollo (Kaasalainen et al. 2007), 54509 YORP (Lowry et al. 2007; Taylor et al. 2007), 1620 Geograph (Durech et al. 2008a), and 3103 Eger (Durech et al. 2012). For example, by the results of numerous high-precision observations of lightcurves obtained in the period from 1980 to 2005, spin rate acceleration for the asteroid 1820 Apollo was experimentally derived (diameter D = 1.4 km, rotation period P =3.065 h, ecliptic coordinates of the pole λ0 = 50°, β0 = 71°). It is dω/dt = (5.3 ± 1.3) × 10−8 radian day−2, which is equivalent of rotation period increment dP/dt = (1.4 ± 0.3) × 10−6 h yr−1 and this value is in good agreement with the theoretically predicted one in the case of the YORP effect influence (Kaasalainen et al. 2007). Taking into account these results, one can suppose that if the influence of the solar radiation (i.e., YORP effect) on rotation of the particular NEAs with D ≥ 1 km in size is enough to be detected from observations, then it is probable that its influence on NEA rotation rate distribution could be noticeable too.
Rossi et al. (2009) carried out numerical modeling of evolution of NEO spin rate distribution under the influence of planetary encounters and YORP effect. Using the Monte Carlo numerical code, they obtained the steady state cumulative distribution of NEO spin rates, which completely reproduces the observed one. The authors concluded that their simulations indicated that the YORP effect was very effective and a dominant mechanism of reshaping spin rate distribution of NEOs from the initial to the currently observed one. They also indicated that planetary encounters alone were not effective enough in reproducing the NEO distribution. As it turned out, in the absence of the YORP effect, the steady state spin rate distribution of NEOs significantly deviates from the observed one.
If that is the case, there can exist some other observational evidence, related to the physical and geometrical characteristics of NEOs, which testifies to the YORP-effect influence on NEO rotation. The purpose of this study is an attempt to reveal them, in spite of the fact that expected effects could be quite negligible, that is, at the threshold of detectability at best.
Spin Rate Distribution of NEAs
Figure 1 shows the spin rate distribution for NEAs and large and small Main Belt asteroids, which were obtained from the Asteroid Lightcurve Database (http://www.minorplanet.info/lightcurvedatabase.html), compiled by A. W. Harris and B. D. Warner (Warner et al. 2009). The distribution of NEAs (Fig. 1a) clearly shows excesses of bodies with fast (ω = 8–11 rev day−1) and slow (ω = 0–2 rev day−1) rotations. Such excesses are just outlined in the distribution of Main Belt asteroids of comparable sizes (D < 5 km) and they are totally absent in the distribution of large ones (Figs. 1b and 1c), which has also been shown in Polishook and Brosch (2009).
It must be emphasized that parameters of rotation of binary NEAs (nearly 40 such systems have already been detected) can actually contribute significantly to formation of these excesses, because according to available data (http://www.johnstonarchive.het/astro), central bodies of such binary systems rotate mainly with periods Pprim = 2.2–3.6 h that are equivalent to ωprim = 7–11 rev day−1 and in that way these objects fall into the excess of fast rotators. In principle, the YORP effect must influence the spin rate of the central (primary) body. However, we did not use the rotation periods of binary near-Earth asteroids, as the axial angular momentum of the primary components could have been turned into the orbital momentum of satellites in case of a recent rotational fission (Walsh et al. 2008; Scheeres 2009; Ćuk and Nesvorný 2010), thus masking the effect of YORP on the rotation period.
Analyzing the rotation rate distributions for MBAs of different sizes and for NEAs (the latter are on average significantly smaller than the smallest MBAs that can be observed today), one can note that the smaller the average size of asteroids of fixed population, the more clearly we can see excesses of fast and slow rotators (see also Polishook and Brosch 2009). Such dependence of rotation rates on asteroid sizes is just typical for the YORP-effect influence. This motivates us to analyze the average size of the NEAs depending on their rotation rate, and to search for correlation in both excesses of the distribution.
Spin Rates and Average Diameters of NEAs
The database “Physical Properties of Near-Earth Objects” by Gerhard Hahn (http://berlinadmin.dlr.de/SGF/earn/nea/), which includes parameters such as absolute magnitude, albedo, diameter, color, taxonomy of asteroids, etc., was used. When the diameter of NEA is unknown due to the absence of information about its albedo, it was calculated by a well-known formula (see, e.g., Tedesco and Veeder 1992):
where D is diameter in km, H is an absolute magnitude, and pv is a geometrical albedo of asteroid. The latter was taken as an average value for the given composition type of asteroid according to Lupishko and Di Martino (1998). Thus, the spin rate data of only those NEAs was used, for which diameters or composition types (taxonomic classes) are known.
Figure 2 presents the obtained dependence. As there is very big data scattering due to the real ranges in measured spin rates of NEAs and in their estimated diameters, we carried out the data smoothing by means of their group averaging, depending on their position in the plot D (km) versus ω (rev day−1). Doing this, we tried to lump the asteroids in excesses and in the middle of distribution together minimally. Such procedure gave us 12 points in the plot (instead of a very scattered 271 points), indicating some possible dependence. The error bars of the points were calculated in a standard way and they point out the mean square error. The points were approximated by the second-degree polynomial equation (solid line in Fig. 2). Dotted lines give the confidence interval corresponding to 0.95 confidence probability. The coefficients of regression of the approximation are given in Table 1. All this also concerns the dependences in Figs. 3 and 4. The obtained dependence shows that average size of NEAs is decreasing from the middle of the distribution of their rotation rates to its ends, that is, the excesses of slow rotators (ω ≤ 2 rev day−1) and fast rotators (ω ≥ 8 rev day−1) are composed by NEAs, which are on average about 1.5 times smaller in size than those in the middle of the distribution. In a qualitative sense, this result is in excellent agreement with the character of influence on spin rate of NEAs just by the YORP effect.
Table 1. Coefficients of regressions and Student's criteria
Linear coefficient of regression
Quadratic coefficient of regression
t2 table value
D (km) – ω (rev day−1)
0.23 ± 0.06
−0.023 ± 0.003
Erel – ω (rev day−1)
−0.11 ± 0.01
0.0013 ± 0.0007
(1 − pv) – ω (rev day−1)
−0.007 ± 0.003
0.0005 ± 0.0002
Spin Rates of NEAs and Their Relative Insolation
As already mentioned, the YORP effect depends also on the amount of solar radiation that the body receives in its orbit, that is, it depends on time-averaged (within one revolution in its orbit) value of inverse squared distance from the object to the Sun. This value is defined by semimajor axis and eccentricity of the orbit. The larger the amount of the energy received, the stronger should be an action of this effect on acceleration or deceleration of spin rate. That is why it is expedient to study possible correlation between rotation rate and amount of the solar radiation that the NEAs receive.
It is enough to calculate the relative amount of the solar energy received from the Sun by each NEA within a single orbital revolution. The formula for calculation of relative insolation Erel is derived under the assumption that for the given asteroid, Erel is proportional to the time-averaged (within a single revolution in the orbit) inverse of the square of heliocentric distance (see also Nesvorný and Vokrouhlický 2007):
where a is a semimajor axis of an asteroid's orbit and e is its eccentricity.
Figure 3 shows the obtained correlation of NEA spin rates and relative values of their insolation. It is seen that asteroids of both excesses are in the orbits where they receive about 10% of solar energy more than in orbits of NEAs, which are in the middle of the distribution. This correlation is less confident than the one in Fig. 2, but it also qualitatively agrees well with the character of the YORP-effect influence and can be considered as another independent argument in favor of it.
Spin Rates and NEAs, Albedos
It is obvious that two identical NEOs in the same orbits (that is, at the same insolations), but with different albedos (e.g., 0.50 for E-type and 0.05 for C-type asteroids), will absorb and re-emit different amount of solar radiation depending on their albedos. In some papers (see, e.g., Breiter et al. 2007; Nesvorný and Vokrouhlický 2007), it is shown that the thermal component of YORP torque is the largest one and for an object with very small surface thermal conductivity, it is proportional to (1 − pv), where pv is the object's albedo. It means that if the excesses of fast and slow rotators in Fig. 1a are really conditioned by the YORP effect, the objects in both excesses should have lower albedos on average than those in the middle of the distribution. Figure 4 shows the obtained result. In spite of the fact that obtained dependence is not strong enough (because of large errors in asteroid albedos), it qualitatively confirms the stated suggestion and therefore this result can be also considered as one more argument in favor of the YORP effect.
Table 1 contains the coefficients of regressions of approximated dependences, given in Figs. 2-4, and corresponding Student's criteria t1 and t2 (calculated and tabulated, respectively). The correlation is significant if t1 > t2. One can see that the obtained correlations are significant, with probabilities indicated in the last column of the table. The correlation between ω (rev day−1) and Erel is significant with probability of only 90%, but the two other correlations are quite significant, indicating that the obtained dependences can be considered real ones.
In this work, we statistically search for correlation between relevant parameters and the YORP effect on rotation rates of NEAs. As a result, the new data (three different dependences), which qualitatively agree well with an influence of the effect on the rotation of NEAs, are obtained. The novelty of these results is that in contrast to published estimates of the YORP-effect influence on spin rate of the four particular NEAs, in this work, all observation data point in favor of the effect's influence on the distribution of spin rates of the whole NEA population. This conclusion is in good agreement with results of Rossi et al.'s (2009) modeling, according to which the YORP effect is a very effective mechanism in reshaping the rotation rate distribution of the NEO population.