tokens: were mainly from the NEOWISE database ( Mainzer et al. 2011 ) with some values taken also from Akari ( Usui et al. 2011 ) and IRAS ( Tedesco et al. 2004 ) catalogs. 4.1. Pole latitude versus size The distribution of the pole latitude β for main-belt asteroids as a function of asteroid size is shown in Fig. 4 . The distribution is strongly bimodal for asteroids smaller than 20–30 km, which can be satisfactorily explained as an effect of a YORP-driven evolution ( Hanuš et al. 2011 , 2013b ). Due to the YORP effect, small asteroids evolve towards the extreme values of obliquity. In the lower panel of Fig. 4 , we show the fraction N p / N total of the number of prograde ( β > 0) rotators in a running box over N total = 100 asteroids as a function of size. For asteroids larger than 100 km, the number of prograde rotators is statistically higher ( N p = 68, N total = 108, probability of the null hypothesis that N p = N total /2 is p ≃ 0.7% assuming binomial distribution) in accordance with the model of Johansen Lacerda (2010) who suggested that the preferentially prograde rotation of large asteroids is a result of accretion of pebbles on planetesimals in a gaseous environment. On the other hand, for asteroids in the size range 1–10 km, there is an excess of retrograde rotators ( N p = 520, N total = 1276, p ≃ 2 × 10 −11 ). For asteroids between 10 and 100 km, the number of prograde and retrograde rotators is statistically the same ( N p = 441, N total = 919, p ≃ 22%). Because most asteroids smaller than about 30 km have large absolute values of β , the prograde/retrograde analysis is not sensitive to asteroids with | β | 30°. The ratio is almost the same even if we restrict ourselves to | β | > 30° where the distinction between prograde and retrograde rotation is unambiguous even for nonzero inclination. This is not true for asteroids larger than 100 km, where for | β | > 30° we have N p = 34, N total = 55, and p ≃ 8%. Fig. 4. The distribution of pole latitudes β for all full and partial models of main-belt asteroids. The bottom panel shows the running-box (over 100 asteroids) fraction of prograde ( β > 0 deg) models (red curve) and also the fraction of models with β > 30 deg (green curve). The dashed and dotted lines show the 5% and 1% significance levels at which the null hypothesis that the prograde and retrograde asteroids are equally frequent can be rejected. The excess of small retrograde rotators in Fig. 4 is statistically significant, however, it is not clear if the reconstructed distribution of β is the same as the real one. Although we are not aware of any bias in the observations or the method that could cause the asymmetry in ± β , there could still be some nontrivial systematic effect that we have not taken into account. 4.2. Pole latitude distribution across the main belt The distribution of pole latitude β is not only dependent on the size, but also on the proper semimajor axis, namely on the proximity to resonances. For asteroids in
ner_tags: O O O O B-Database I-Database O B-Citation I-Citation I-Citation I-Citation O O O O O O O B-Telescope O B-Citation I-Citation I-Citation I-Citation O O B-Telescope O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O