9 Concluding remarks

The comparisons presented in the last section show that the coefficients of high frequency nutations or of corresponding polar motions, as presented by different groups, do not differ by more than a few tenths of a microarcsecond for any of the frequencies involved. This is not surprising, considering that the largest of the polar motion amplitudes are under 20 $\mu$as. We focus here, therefore, on a few special features referred to in earlier sections.

Firstly, the Chandler resonance in the low frequency polar motions: We pointed out at the beginning of Sect. 6 that the resonance frequency varies with the frequency of excitation. We find that the largest effect of this variation is on the $\pm$3231 day polar motions: -0.08 and 0.16 $\mu$as, respectively, on the coefficients $x_{\rm p}^{\rm s}$ and $x_{\rm p}^{\rm c}$ of the retrograde one, and -0.17 and 0.02 on those of the prograde one. These are not ignorable at the 0.1 $\mu$as level. Another point concerns the flipping of the sign of the imaginary part of the resonance frequency which has to accompany the passage from positive to negative excitation frequencies. If this flip were ignored, $(x_{\rm p}^{\rm s}, x_{\rm p}^{\rm c})$ would be in error by ( -0.17, 0.02) $\mu$as for the -3231 day polar motion.

Secondly, the presence of the secular term in the polar motion of the nonrigid Earth due to the constant part of the (4,0) potential (Sect. 6 and Table 4), which was noted already by Mathews & Bretagnon (2002) and Brzezinski & Capitaine (2002): It seems necessary to point out here that the "observed secular motion of the pole'', referred to by the latter, is a linear drift of the Earth's rotation pole in relation to the TRF while the polar motions dealt with here (and elsewhere) in the context of the transformation between the CRF and the TRF are the secular and periodic motions (with periods outside the retrograde diurnal band) of the pole of the CRF as seen from the TRF. It is unfortunate that the use of the term "polar motion'' with different meanings in different contexts lends scope for avoidable confusion.

Thirdly, the effects of possible triaxiality $Z_{\rm f}$ of the core: We have brought out explicitly the coupling of the retrograde and prograde diurnal wobbles that is produced by the Z and $Z_{\rm f}$terms in the angular momentum, and the role played thereby in the prograde semidiurnal nutations by the FCN resonance (more specifically, by the resonance in the retrograde diurnal wobble of the core) through this coupling. We find the increments to the wobble eigenfrequencies due to triaxiality to be of no observational consequence: they are of the second order in the triaxiality parameter, proportional to $e'^2\equiv ZZ^*$ (as noted in the first para of Sect. 7.3) or to $e'_{\rm f}{}^2\equiv Z_{\rm f}Z_{\rm f}^*$, as the case may be, in contrast to the first order changes found by Escapa et al. (2002). (It is hard to comprehend how their first order changes could involve the moduli of the triaxiality parametes rather than the complex parameters themselves.) In regard to the forced motions, we have confirmed that the contributions from core triaxiality $Z_{\rm f}$ could be significant, as observed by Escapa et al. especially if $Z_{\rm f}$ were a few times Z; but the numbers we obtain for the these contributions are only about 40% of those reported by these authors, or less. Brzezinsky & Capitaine (2002) have computed $Z_{\rm f}$ from the mantle tomographic model of Morelli & Dziewonski (1987) and the mantle convection model of Defraigne et al. (1996). But the modeling of the core mantle boundary with the precision needed to make a useful estimate of $Z_{\rm f}$ is notoriously difficult, and the general consensus seems to be that the value of $Z_{\rm f}$ remains highy uncertain. One might perhaps hope that, if the core triaxiality were large enough, its estimation from observations of diurnal polar motions/semidiurnal nutations would become possible if and when the precision of estimation of such motions approaches the 1 $\mu$as level. However, it is important to take note that the prograde diurnal polar motions considered here, which are due to the torques exerted on the triaxial structure by degree 2 tesseral (m=1) tidal potentials, are at least 10 times smaller than those due to the ocean tides raised by the very same tidal potentials; see Table 6 of Chao et al. (1996) for a comparative listing of observational estimates and theoretical predictions from various works. Even if the contribution from $Z_{\rm f}$, which is largest for the nutation period of 0.49863 days (0.99727 day polar motion), were as large as a few $\mu$as, it would still be only a few percent of the ocean tide contribution for the same period. To estimate the core triaxiality contribution accurately enough from observations on prograde diurnal polar motions to permit useful bounds to be placed on the triaxiality $Z_{\rm f}$, one needs predictions for the dominant ocean tide contribution that are good at least at the 1 $\mu$as level. The prospects for modeling the ocean tide contribution to this level of accuracy seem quite dim, given the uncertainties in the modeling of ocean tides and their effects.

Acknowledgements

We are happy to acknowledge illuminating discussions with Aleksander Brzezinsky during the final stages of the preparation of this paper. We are also indebted to an anonymous referee for a detailed and thoughtful review of the paper, leading to improvements in presentation.