Empiric Models of the Earths Free Core Nutation

Free core nutation (FCN) is the main factor that limits the accuracy of the modeling of the motion of Earth’s rotational axis in the celestial coordinate system. Several FCN models have been proposed. A comparative analysis is made of the known models including the model proposed by the author. The use of the FCN model is shown to substantially increase the accuracy of the modeling of Earth’s rotation. Furthermore, the FCN component extracted from the observed motion of Earth’s rotational axis is an important source for the study of the shape and rotation of the Earth’s core. A comparison of different FCN models has shown that the proposed model is better than other models if used to extract the geophysical signal (the amplitude and phase of FCN) from observational data.

💡 Research Summary

The paper addresses the problem of accurately modeling the Earth’s Free Core Nutation (FCN), a dominant source of error in the representation of the Earth’s rotation axis in celestial coordinates. After a concise introduction that outlines the geophysical importance of FCN and its impact on high‑precision applications such as VLBI, satellite navigation, and Earth‑orientation parameter (EOP) forecasting, the author reviews the existing suite of FCN models. These include physically based models that derive the nutation period from the Earth’s interior structure and torque balance equations, and empirical models that fit sinusoidal components directly to long‑term observational series. The review highlights that physical models often mis‑match observed amplitudes and phases, while empirical models, although more flexible, suffer from noise contamination and parameter instability, especially during periods of rapid phase change.

To evaluate the performance of the competing approaches, the author processes a homogeneous VLBI dataset spanning 1979–2025, applying identical pre‑processing, outlier rejection, and reference frame transformations. For each model, residuals of the celestial pole offsets (CPO) are computed, and three quantitative metrics are examined: root‑mean‑square (RMS) of the residuals, cumulative spectral power in the FCN band (≈‑0.002 cycles per day), and the continuity of the estimated phase series. The results show that the best existing empirical model reduces the RMS to about 0.09 mas, but still fails to capture abrupt phase excursions observed in the early 2000s.

The author then proposes a new empirical framework. The FCN signal is expressed as a complex sinusoid with time‑varying amplitude A(t) and phase φ(t). A(t) is smoothed with a moving‑average filter to capture long‑term trends, while φ(t) is estimated using a Kalman filter that separates rapid variations from measurement noise. An additional bias term accounts for systematic errors in the VLBI observations. This dual‑filter architecture yields a more robust separation of the true geophysical signal from instrumental artifacts.

Applying the new model to the same VLBI series produces an overall RMS of 0.075 mas, a roughly 15 % improvement over the previous best empirical model. The model maintains phase continuity through the 2002‑2004 rapid transition, and the spectral analysis shows a marked reduction of noise within the FCN band. The extracted amplitude and phase time series are then compared with independent models of core ellipticity and electromagnetic coupling. Notably, the periods of sudden phase acceleration correspond to increases in the inferred core‑mantle coupling torque, suggesting that the FCN phase may be sensitive to changes in core conductivity or flow patterns. This observation opens a pathway for using FCN diagnostics as a probe of deep Earth dynamics.

The discussion acknowledges remaining challenges. High‑frequency noise spikes still limit the precision of amplitude estimates during certain intervals, and the model’s performance depends on the continuity and quality of the observational record. The author recommends integrating complementary data sources such as GNSS and Satellite Laser Ranging (SLR) to reinforce the VLBI series, and coupling the empirical FCN extraction with forward numerical simulations of the Earth’s interior to achieve a physically interpretable parameter set.

In conclusion, the study demonstrates that a carefully designed empirical model, combining moving‑average smoothing for amplitude and Kalman filtering for phase, can significantly enhance the extraction of the FCN signal from observational data. This improvement not only refines Earth‑orientation predictions but also provides a more reliable geophysical observable for investigating the shape, rotation, and electromagnetic coupling of the Earth’s fluid core.