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tokens: this value will be subject to a selection bias as it is easier to obtain unambiguous light curves for principal axis rotators, it is unlikely that a debiased value would reach the 40 per cent that our modelling suggests is required to explain our data. Therefore, we conclude that none of the above mechanisms can currently explain our observed trends with size. 5.2 Comparing results for the inner and outer main belt populations Figs 3 and 4 show that the rate at which the distribution of magnitude variation changes with diameter is greater for the inner belt than the outer belt. This may be due to either a difference in the reorientation of the objects between the two populations, or a difference in shape distribution. Unlike the difference in variability with diameter, the difference with semimajor axis is potentially more straightforward. Reorientation due to YORP will happen more slowly at greater heliocentric distances and thus objects in the outer belt should show systematically less variability, all other factors being equal. This agrees with what we see here. To test this, we assumed that the shape distribution of the outer belt was identical to that obtained for the inner belt. In order to assess the validity of this assumption, a best-fitting truncated Gaussian shape distribution was obtained for the outer main belt independently at 2 D 3 km using the previously outlined method. This gave an average axial ratio of 1 : 0.87 ± 0.09 : 0.74 ± 0.11 corresponding to a population with a shape distribution μ = 0.90, σ = 0.11. This is not significantly different from that found for the inner belt and therefore our assumption holds. The difference between the two populations must therefore be due to a difference in the degree of reorientation. Part of this may be due to the slower rate of YORP in the outer belt. At present our statistical model does not account for these effects and this may offer an avenue for future work. 5.3 Previous constraints on shape and spin A previous study by Szabó Kiss ( 2008 ) used a similar sparse light-curve sampling method, utilizing data from the SDSS Moving Object Catalogue. Assuming all objects to have spin axis parallel to the sky plane, they reported a shape distribution in good agreement with axial ratios from then-published light curves. We compared our findings to an updated distribution of axial ratios from current light curves (Kryszczyńska et al. 2007 ). The light curves given in this data base suggest an average asteroid shape with 1σ spread with axial ratio 1 : 0.76 ± 0.13 : 0.62 ± 0.11. This result is within 1σ of our own findings, albeit both 〈 b / a 〉 and 〈 c / b 〉 are smaller than our results. We then used the truncated Gaussian distribution for b / a and the approximated distribution for b / c obtained from the Kryszczyńska et al. ( 2007 ) data set, and varied the upper obliquity limit in an attempt to obtain a good fit from this large diameter shape distribution. It was possible to obtain