Report of IAU Working Group on ‘Non-Rigid Earth Nutation Theory’

With respect to the publication in the Proceedings of the IAU Colloquium 180 (Dehant 2000), I have a few additional remarks that I would like to make. There were three competing models:1. MHB2000 of Mathews et al. (2000), (see also Mathews 2000), constructed from a rigid Earth nutation series with about 1400 terms and a transfer function obeying the sum rules, based on the fit of a semianalytical nutation theory to VLBI observations, with resonances determined by geophysical parameters including those estimated from the fit, incorporating the atmospheric annual effect fitted to observations, including ocean tide effects based on admittances with frequency dependence due to the FCN (Free Core Nutation) resonance and to ocean dynamics (fitted to ocean tide data), including electromagnetic couplings of the fluid core, and with mutual consistency maintained in the treatments of nutations, solid Earth tides and ocean tides which influence one another;2. GF2000 of Getino and Ferrándiz (2000), using a global Hamiltonian approach for 106 waves fitted to VLBI data, incorporating a resonance with global parameters fitted to observations such as the FCN free mode frequency, compliances, and dissipation coefficients, incorporating ocean corrections from Huang et al. (2000) with a frequency dependent resonance and fitted atmospheric corrections, and necessitating empirical corrections;3. SF2000 of Shirai and Fukushima (2000a and 2000b) or Herring (2000, not published), empirical models, based on a simple resonance formula fitted to the VLBI observation.

Comparison of Nutation Theories

Analyses of residuals between VLBI observations and combinations of nutation series show that the MHB 2000 nonrigid-Earth nutation model applied to the REN 2000 rigid-Earth model results in the best fit, and that amplitudes of any possible periodic terms remaining in the observed corrections to the MHB2000 theory could be expected to be less than 0.1 mas.

A New Nutation Model Of Nonrigid Earth With Ocean And Atmosphere

By integrating the truncated complex scalar gravitational motion equations for an anelastic, rotating, slightly elliptical Earth, the complex frequency dependent Earth transfer functions are computed directly. Unlike the conventional method, the effects of both oceanic loads and tidal currents are included via outer surface boundary conditions, all of which are expanded to second order in ellipticity. A modified ellipticity profile in second order accuracy for the non-hydrostatic Earth is obtained from Clairaut’s equation and the PREM Earth model by adjusting both the ellipticity of the core-mantle boundary and the global dynamical ellipticity to modern observations. The effects of different Earth models, anelastic models, and ocean models are computed and compared. The atmospheric contributions to prograde annual, retrograde annual and retrograde semiannual nutation are also included as oceanic effects. Finally, a complete new nutation series of more than 340 periods, including in-phase and out-of-phase parts of longitude and obliquity terms, for a more realistic Earth, is obtained and compared with other available nutation series and observations.

Improved Models for Precession and Nutation

The modeling of nutations and precession has advanced to the point where the rms of residuals between theory and the observational estimates from the VLBI data of the past decade is only 0.16 mas in Δψ sin є0 as well as in Δє. Such a fit is provided by the MHB2000 nutation series (Mathews et al., 2000) based on geophysical theory with a few basic Earth parameters estimated by a fit to nutation-precession data, and its accompanying precession rate. A brief account of the series is presented, along with an outline of the theoretical background and of the geophysical information of interest obtained in the process of constructing the series. A series due to Shirai and Fukushima (2000) also gives a somewhat comparable fit to data, improving on the IERS 1996 series, but it is essentially empirical and provides no geophysical insights.

Improvement of Nonrigid-Earth Nutation Theory by Adding a Model Free Core Nutation Term

From the analysis of VLBI observational data compiled by USNO (U.S. Naval Observatory) from MJD 44089.994 to 51618.250 (McCarthy, 2000), we showed that a strong peak around –400 sidereal days in the spectrum of its differences from the IERS96 nutation theory could be explained by adding a model Free Core Nutation (FCN) term in the form of a single damped oscillation. Then we developed a new analytical theory of the nonrigid-Earth nutation including the derived FCN model. We adopted RDAN98 (Roosbeek and Dehant, 1998) as the rigid Earth nutation theory. It was convolved with a transfer function using numerical convolution in the time domain (Shirai and Fukushima, 2000). The form of the transfer function was the same as that of Herring (1995). However, its free parameters such as the complex amplitude and frequency of the FCN were readjusted by fitting to the above VLBI data. Even after truncating the forced nutation series so as to contain only 180 terms, the WRMS (Weighted Root Mean Square) of the complex residuals for the new nutation series is 0.312 mas, which is significantly smaller than 0.325 mas, that of the IERS96 nutation theory. As for the FCN term, we estimated its oscillatory period as –430.8±0.6 sidereal days, and its Q-value as 16200 ± 1600. Also we estimated the correction of the precession constants as −0.29297±0.00047”/cy in longitude and −0.02430±0.00019”/cy in obliquity, respectively.

Report of the IAU Working Group on ‘Nonrigid-Earth Nutation Theory’

The last precession-nutation model adopted by the IAU (International Astronomical Union) in 1980 is the nutation series built on Wahr’s Earth transfer function for the nutations of an oceanless elastic Earth (Wahr, 1979, 1981), and on Kinoshita’s rigid-Earth precession-nutation series (Kinoshita, 1977; Kinoshita et al., 1979). The resulting precession and nonrigid-Earth nutation series have been used since that time and have been compared with observations. This comparison, which has been done by different teams all-over the world, shows that the theoretical series must be improved to meet observational precision. A Working Group (WG) was set up to examine the possibility of adopting a new nonrigid-Earth nutation series and to study the existing possibilities. On the one hand, the rigid-Earth nutation series have been improved (three new series) and the mutual differences have been shown to be less than a few hundreds of microarcseconds. On the other hand, new Earth transfer functions have been derived based on additional physical considerations within the Earth. The problem with these transfer functions however is that there is no reliable, independent information about the geophysical parameters needed to improve a theoretical model. Instead, the discrepancies with the nutation observations themselves are used to infer those parameters. Recent fits of geophysical parameters to the observed nutations have provided a series that is suitable for practical use, and is also a source of important information on the physics of the Earth’s interior. This paper reviews the recent work of the WG and establishes the reasons and criteria for the choice of the new model ‘IAU 2000’ which is proposed for adoption at the next IAU General Assembly.

Towards sub-microarsecond rigid Earth nutation series in the Hamiltonian theory

The nonrigid Earth nutation series adopted by the IAU (International Astronomical Union) are based on the works of Kinoshita (1977) and Wahr (1979). In the first one, the rigid Earth nutation series were calculated by the application of the Hamiltonian canonical equations to the rotation of the rigid and elliptical Earth. In the second one, the transfer function for the nutations of an elliptical, elastic and oceanless Earth with fluid core and a solid inner core was obtained. The nonrigid Earth nutation coefficients were derived from the convolution between Wahr’s transfer function and Kinoshita’s rigid Earth nutation series.The improvement in the accuracy of the techniques as a Very Long Baseline (VLBI), Lunar Laser Ranging (LLR) and Global Positioning System (GPS) has led in this decade to the extension of Kinoshita’s theory and more precise determination of Wahr’s transfer function. In the present paper and starting from Kinoshita’s work (1977), we present the different steps carried out, during this last decade, to obtain the sub-microarcsecond rigid Earth nutation series REN 2000 from the Hamiltonian study of the rotation of a rigid Earth (Souchay et al., 1999).