tokens: dependent on the grain shape or composition. Indices fall between −2.4 and −2.74, which gives slightly less steep fits than in the analysis in LH07 , where a spectral index −3 provides the best fit for a severely more limited set of shapes. Therefore, while RAT scaling predictions can be done with a simple power-law estimate, our study suggests that a power-law index −2.6 should be used for a general fit. Figure 20. Q Γ (Θ = 0) as a function of λ / a eff for the silicate base shapes. The shaded area around the mean indicates the range of all samples. The power-law fit is done for data between λ / a eff = 2 and 20. Figure 21. Same as Figure 20 , but for coated silicates. Figure 22. Same as Figure 20 , but for carbonaceous shapes. Due to the limited wavelength range available, Figures 20 – 22 cannot show whether or not the RAT efficiency is best fitted by a constant below λ / a eff = 1.8, an estimate of the threshold between constant and power-law fits that originates from LH07 . The region λ ∼ a eff is not possible to account for properly using the limited wavelength range. Due to this, the full RAT fit provided in LH07 cannot be confirmed for the ensembles. Finally, we consider the precession-inducing RAT component Q e 3 . From LH07 , we know how to estimate Q e 3 for the AMO spheroid, here chosen to have an aspect ratio a : b = 2:3, aswhere e is the spheroid eccentricity, a is its minor axis, and K is a fitting factor. For rough estimates of the maximum value , which are found at cos Θ = ±2 −1/2 , LH07 showed that K is near unity. We study the wavelength-dependent normalized values of for the silicate oblate and prolate ensembles. Using Equations ( 3 ) and ( 4 ) with K ≈ 1, we also study the deviation from the estimated functional form of AMO. The results, presented in Figure 23 , reveal that there exists a critical value of λ / a eff ≈ 5, where the maximal value and shape deviation against AMO of Q e 3 are clearly anticorrelated, such that the maximum value of Q e 3 is noticeably less and the shape is considerably different than what AMO predicts. Such a clear discrepancy against AMO was identified to be a property of only the Q e 3 component. Above λ / a eff ≈ 5, quickly peaks, after which it gradually decreases. Also, below the critical value, another strong decrease is visible, although it is limited by the wavelength range used. That behavior, which is almost identical for the two different shape ensembles, can possibly be identified as being due to changing scattering regimes. To fully understand the universality of the critical grain size, which implies that in some situations certain grain size groups may exhibit a different types of alignment, further studies are needed. Figure 23. Top panels: mean values of the component, normalized with the corresponding AMO component, as a function of λ / a eff , for the silicate oblate and prolate ensembles. The bars represent 1 standard deviation around the mean. Below: mean deviation from AMO
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