Abstract. Variability near a 6-yr period has been reported in the length of day, motions within the Earth’s fluid core, several climatic parameters, and atmospheric angular momentum. Here we demonstrate the robustness of a quasi-6-yr oscillation in atmospheric angular momentum using several independent atmospheric reanalysis products over 1980–2020. This signal is highly significant, consistent across datasets, and accounts for up to about 25 % of atmospheric angular momentum variance at interannual time scales. Its expression in the atmospheric zonal wind circulation exhibits a coherent vertical structure throughout the troposphere, with maximum amplitudes near the tropopause in the tropical belt. In addition, the 6-yr oscillation in zonal winds is in phase across from southern to northern latitudes. This structure distinguishes the 6-yr signal from the annual cycle and from ENSO-related variability, and points to a large-scale, organized component of the atmospheric circulation, consistent with alternating phases of weaker and stronger atmospheric super-rotation relative to the solid Earth. While the origin of the length-of-day 6-yr cycle is relatively well established and attributed to exchange of angular momentum from the core to the mantle, the process underlying the 6-yr variability in the zonal wind circulation remains to be elucidated.

In this study, we revisit Kaula’s orbit perturbation theory to evaluate the capability of satellite orbits to sense degree-1 gravitational signatures. We explicitly derive the Kaula-based expressions for the degree-1 potential terms and compute the corresponding inclination F1mp(i), and eccentricity functions G1mp(e), providing—for the first time—complete tables for degree-1.As a study case, we applied this to the ESA's Genesis mission set to be launched in 2029. Genesis will be the first in orbit geodetic observatory carrying onboard the 4 co-located geodetic techniques (GNSS, SLR, DORIS and VLBI). The dependence on orbital inclination of the expressions for the degree-1 potential coefficients V10 and V11 is examined. Our analysis shows that V10, associated with the geocenter’s z-component, scales with sin i, implying that polar orbits maximize sensitivity to vertical geocenter variations, while equatorial orbits remain largely insensitive to them. Conversely, the two-term structure of V11 enhances the detectability of the geocenter’s x- and y-components for equatorial or near-equatorial orbits, with reduced but non-negligible sensitivity in polar configurations.This work demonstrates how Kaula’s theory can guide the design of future gravimetry missions by identifying orbital parameters that optimize degree-1 recovery, thereby improving geocenter estimation and strengthening the link between satellite gravimetry and terrestrial reference frame realization.

Dynamic loads in planetary mantles have the potential to deform the core-mantle boundary (CMB). On Earth, subducting slabs primarily induce a degree 2-order 2 deformation of the CMB in the spherical harmonic (SH) reference system. On Mars, the presence of the dichotomy and of the Tharsis region could produce loading across multiple degrees and orders, including degree-1, degree 2-order 2, degree 2-order 0, and degree 3-order 3 components. Thanks to the InSight (Interior exploration using Seismic Investigations, Geodesy, and Heat Transport) mission's radio science experiment, observations of Mars' nutations are now available. Periodic length-of-day (LOD) variations of Mars have been detected first by radio tracking the Viking landers, and InSight data have indicated the presence of a secular trend in LOD. In the case of nutations, the Martian core's non-hydrostatic flattening plays a first-order role in determining nutation amplitudes. In this study, we explore second-order effects arising from dynamic topography at the CMB. We compute the pressure exerted on the CMB topography inside Mars' liquid core and evaluate the resulting topographic pressure torque acting on the boundary, which can influence both nutations and LOD variations. Our results show that, albeit at microarcsecond (& micro;arcsec) level - well below current observational thresholds, the most significant contribution to nutations arises from degree 2-order 2 component. As for LOD variations, while Earth exhibits notable contributions from inertial wave resonances, the situation on Mars is different. The planet's tidal LOD variations have periods that are either too long or too far apart from those of inertial waves. Consequently, the associated contributions fall below the level of detectability.

We present updated estimates of Basic Earth Parameters (BEP) from VLBI Celestial Pole Offset (CPO) time series spanning 1980-2025 using an ensemble Markov Chain Monte Carlo (MCMC) Bayesian inversion. Building upon , we incorporate recent advances in ocean tidal modeling and update several aspects of the algorithm. Key improvements include: (1) implementation of a cubic spline representation for Free Core Nutation (FCN) amplitude variations, which significantly reduces multimodality in FCN-related parameter in MCMC sampling compared to a linear representation; (2) integration of updated Ocean Tidal Angular Momentum (OTAM) values from FES2014 ocean tidal atlas , without the empirical 0.7 scaling factor previously applied in the construction of the last adopted nutation model MHB2000; and (3) utilization of five diverse CPO series from different analysis centers spanning up to 45 years of observations. Our estimated mean values of the parameters show good consistency across different CPO series, with values of the Earth's dynamical ellipticity at the edge of the 1 sigma range of MHB2000 . Notable findings include a larger absolute value for the imaginary part of the core-mantle boundary (CMB) coupling constant ( K CMB ), approaching the 2 sigma boundary of , which may reflect contributions from other coupling mechanisms in addition to electromagnetic coupling, including possible topographic coupling through "form drag" effect caused by wave interactions with irregular boundaries . The real part of the Inner Core Boundary (ICB) coupling constant ( K ICB ) is approximately half the MHB2000 value, potentially indicating the need to revisit hydrostatic assumptions for the inner core given recent seismic evidence of viscous deformation . Compliance estimates suggest that frequency extrapolation methods from seismic to nutation bands may be considered unchanged in the mantle, but should be revised at the ICB. The enhanced FCN free-mode modeling captures amplitude variations that differ from empirical models, particularly after 2000, although the physical interpretation of these differences requires further investigation. The systematic discrepancies across multiple parameters suggest that current nutation theory needs substantial updates to incorporate more realistic models of core-mantle coupling and inner-core-outer-core coupling.

Evidence from seismic studies, mineral physics, thermal evolution models and geomagnetic observations is inconclusive about the presence of a stably stratified layer at the top of the Earth's fluid outer core. Such a convectively stable layer could have a strong influence on the internal fluid waves propagating underneath the core-mantle boundary (CMB) that are used to probe the outermost region of the core through the wave interaction with the geomagnetic field and the rotation of the mantle. Here, we numerically investigate the effect of a top stable layer on the outer core fluid waves by calculating the eigenmodes in a neutrally stratified sphere permeated by a magnetic field with and without a top stable layer. We use a numerical model, assuming a flow with an m-fold azimuthal symmetry, that allows for radial motions across the lower boundary of the stable layer and angular momentum exchanges across the CMB through viscous and electromagnetic coupling. On interannual timescales, we find torsional Alfv & eacute;n waves that are only marginally affected by weak to moderate stratification strength in the outer layer. At decadal timescales similarly weak stable layers promote the appearance of waves that propagate primarily within the stable layer itself and resemble Magneto-Archimedes-Coriolis (MAC) waves, even though they interact with the adiabatic fluid core below. These waves can exert viscous and electromagnetic torques on the mantle that are several orders of magnitude larger than those in the neutrally stratified case.

The core is the deepest part of the planets. It is partially or totally liquid in all the terrestrial planets of the Solar System and sometimes generates a magnetic field (it is the case for the Earth and Mercury). Rapidly rotating planets like Earth and Mars, which are in addition inclined in space, undergo gravitational effects from the Sun and their moons (as well as from the other planets to a minor extend). Consequently, Mars and the Earth wobble in space: their rotation axis is doing a precession around the perpendicular to the ecliptic and additional periodic motions called nutations. Nutations provide information about the core and about the coupling mechanisms between the core and the mantle. Additionally, these two planets exhibit variations in their rotation called length-of-day (LOD) variations. The most recent data from using the VLBI (Very Long Baseline Interferometry) technique for the Earth and from the NASA InSight (Interior exploration using Seismic Investigations, Geodesy and Heat Transport) mission on Mars provide insights on the processes modifying their rotation and orientation in space. While nutations are very well understood, LOD are difficult to predict. Seasonal changes are mostly related to external geophysical fluids: atmosphere, ocean, and hydrosphere for the Earth, atmosphere and icecaps for Mars. While we have no precise series long enough for Mars, it is not the case for the Earth for which long timescale (decadal) LOD variations are mostly linked to the core. Even polar motion of the Earth can provide information about the core.

Creating an absolute space-tie where all the geodetic methods are onboard is the key for an improved and stable terrestrial reference frame as well as with various scientific applications. Such satellite concepts have already been proposed to achieve an accurate and stable terrestrial reference frame. Next generation Galileo satellites can provide a single well-calibrated platform for the colocation of the space-based geodetic techniques establishing precise and stable ties between the key geodetic techniques. One of the most crucial and novel aspect of such concepts is the VLBI transmitter (VT) which will emit quasar-like signals from the space to be observed by the VLBI ground stations. VT can directly link the terrestrial and celestial reference frames and bring the unique features of VLBI technique to an Earth orbiting satellite. In the context of call for future Galileo payloads a novel VT has been under development. VT shall be compatible both with the legacy and VGOS antennas. Here, we present the progress on ongoing ESA study for VT for Galileo as well as for other future missions.

A 6-year cycle has long been recognized to influence the Earth’s rotation, the internal magnetic field and motions in the fluid Earth’s core. Recent observations have revealed that a 6-year cycle also affects the angular momentum of the atmosphere and several climatic parameters, including global mean sea level rise, precipitation, land hydrology, Arctic surface temperature, ocean heat content and natural climate modes. In this review, we first present observational evidences supporting the existence of a 6-year cycle in the Earth system, from its deep interior to the climate system. We then explore potential links between the Earth’s core, mantle and atmosphere that might explain the observations, and investigate various mechanisms that could drive the observed 6-year oscillation throughout the whole Earth system.

Having a Very Long Baseline Interferometry (VLBI) transmitter (VT) onboard Galileo satellite allows us to determine the misorientation between GNSS and VLBI frames. To exploit the maximum performance, we study the operational strategies for VLBI ground segment. We simulate VLBI observations of a VT onboard a Galileo satellite to evaluate the rotation transformation between the VLBI and GNSS frames. The contribution of a VT as space tie is assessed by the evaluation of the formal precision of the orientation parameters between the VLBI and GNSS frames using different ground stations/baselines, aiming to find the optimal observation geometry for the best precision on the rotation transformation.

Recent studies have identified a quasi six-year oscillation (QSYO) spanning various Earth system parameters, including the length of day, polar motion, secular variation of the magnetic field, sea level, precipitation, terrestrial water storage, land ice, and winds. This oscillation is linked to fluid dynamics processes in the liquid outer core, yet the mechanism facilitating its transmission from the core to the broader Earth system remains unclear. One of the proposed scenario is the plausible role of the magnetic field as a conduit for transmitting the QSYO from the core to the climatic system Leveraging data from atmospheric and magnetic models alongside observations from Global Navigation Satellite System (GNSS), our approach employs statistical analysis to explore the presence of the QSYO in the atmosphere. We specifically analyze various parameters in both the charged (e.g., Total Electron Content - TEC) and neutral (e.g., nebulosities) atmospheric layers. This study aims to establish the existence of the QSYO in the atmosphere and link its variations with those from core in the magnetic field.

To understand the processes involved in the deep interior of the Earth and explaining its evolution, in particular the dynamics of the Earth’s fluid iron-rich outer core, only indirect satellite and ground observations are available. They each provide invaluable information about the core flow but are incomplete on their own: - The time dependent magnetic field, originating mainly within the core, can be used to infer the motions of the fluid at the top of the core on decadal and subdecadal time scales. - The time dependent gravity field variations that reflect changes in the mass distribution within the Earth and at its surface occur on a broad range of time scales. Decadal and interannual variations include the signature of the flow inside the core, though they are largely dominated by surface contributions related to the global water cycle and climate-driven land ice loss. - Earth rotation changes (or variations in the length of the day) also occur on these time scales, and are largely related to the core fluid motions through exchange of angular momentum between the core and the mantle at the core-mantle boundary. Here, we present the main activities proposed in the frame of the GRACEFUL ERC project, which aims to combine information about the core deduced from the gravity field, from the magnetic field and from the Earth rotation in synergy, in order to examine in unprecedented depth the dynamical processes occurring inside the core and at the core-mantle boundary.

We study coupling mechanisms at the core-mantle boundary (CMB) of the Earth in the frame of nutations and Length-of-Day (LOD) variations. The CMB is usually considered to have a smooth spherical or elliptical shape inducing a Poincaré flow in the nutation case and a global rotation in the LOD case. However, in reality, the CMB is bumpy and there are mountains and valleys representing local height differences of the order of a kilometer. The existence of a topography induces inertial waves that need to be considered in the flow of the core. This is in addition to the Poincaré fluid motion when the nutations are computed and in addition to a relative rotation of the fluid opposite and of the same amplitude as that of the mantle for LOD variations. The additional pressure and the topographic torque depend on the shape of the CMB and can be related to the spherical harmonic coefficients of the CMB topography. We follow the philosophy of the computation of Wu and Wahr [Geophys. J. Int., 128(1), 18-42, 1997] and determine the coefficients of the velocity field in the core at the CMB in terms of the topography coefficients. We used an analytical approach instead of a numerical one. We confirm that some topography coefficients may enhance length-of-day variations and nutations at selected frequencies, and show that these increased rotation variations and nutations are due to resonance effects with inertial waves in the incremental core flow. While they could be at a detectable level for LOD, they are very small for nutations (except for the flattening of the core), enhancing the importance of the electromagnetic coupling at the CMB.

Abstract. Dynamic loads in planetary mantles have the potential to deform the core-mantle boundary (CMB). On Earth, subducting slabs primarily induce a degree 2–order 2 deformation of the CMB in the spherical harmonic (SH) reference system. On Mars, the presence of the dichotomy and of the Tharsis region could produce loading across multiple degrees and orders, including degree-1, degree 2–order 2, degree 2–order 0, and degree 3–order 3 components. Thanks to the InSight (Interior exploration using Seismic Investigations, Geodesy, and Heat Transport) mission’s radio science experiment, observations of Mars' nutations are now available. Periodic length-of-day (LOD) variations of Mars have been detected first by radio tracking the Viking landers, and InSight data have indicated the presence of a secular trend in LOD. In the case of nutations, the Martian core’s non-hydrostatic flattening plays a first-order role in determining nutation amplitudes. In this study, we explore second-order effects arising from dynamic topography at the CMB. We compute the pressure exerted on the CMB topography inside Mars' liquid core and evaluate the resulting topographic pressure torque acting on the boundary, which can influence both nutations and LOD variations. Our results show that, albeit at microarcsecond level—well below current observational thresholds, the most significant contribution to nutations arises from degree 2–order 2 component. As for LOD variations, while Earth exhibits notable contributions from inertial wave resonances, the situation on Mars is different. The planet’s tidal LOD variations have periods that are either too long or too far apart from those of inertial waves. Consequently, the associated contributions fall below the level of detectability.

Each geodetic technique realizes its specific reference frame and traditionally these independent frames are linked through the local-ties at collocation sites. Space ties onboard of a spacecraft can be also used to connect different frames as another complementary approach. Observations of a broadband VLBI transmitter (VT) onboard Galileo satellite by a VLBI ground station network can create such a link between GNSS and VLBI reference frames. In this study, we simulate observations of a VT onboard Galileo satellites and investigate the effect of VT observing network/station selection on the rotation transformation parameters between the VLBI and GNSS frames. We find that the rotation transformation uncertainties can be inflated by more than 25% when a geometrically critical station is removed from the network of 21 ground stations. We additionally analyze various VLBI networks consisting of 8-9 ground stations dedicated to IVS R1/R4 sessions. We report that by adding individual stations into some networks uncertainties are reduced by up to a factor of three in daily solutions and up to 50% for all orientation components when combining solutions covering a Galileo repeat cycle of ten days.

The GRACEFUL project aims to better understand dynamic processes in the Earth’s deep interior using a combination of satellite observations of the Earth’s magnetic field, gravity field, and rotation. Significant oscillations at periods ranging from 6 to 8 years have been linked to dynamical processes in the fluid outer core and at the core-mantle boundary, influencing the three aforementioned observables. However, the occurrence of a 6-year cycle was also recently evidenced in the climate system, expressed in the global mean surface temperature, zonal winds, precipitation, terrestrial water storage changes, sea level changes, and ice mass changes. This study provides a comprehensive review of the 6-year cycle observed across the entire Earth system, from its deep interior to its fluid external envelopes. We propose several mechanisms that could explain small amplitude variations in geodetic observations, such as the length of day and gravity field, highlighting potential links between internal and external geodynamics at periods of around 6 years.

Abstract Observation of rotation variations and tides provides constraints on the interior properties of celestial bodies. Both can be precisely measured with a 6DoF (Degrees of Freedom) motion sensor placed on their surface. This type of instrument measures rotation rates and linear accelerations in a large frequency band, which includes the frequencies involved in the tides and rotation variations. A novel sensor under development aims to measure rates and accelerations with an amplitude spectral density of $${2}\,\upmu \textrm{rad}\, \textrm{s}^{-1} \textrm{H}\textrm{z}^{-1/2}$$ 2 μ rad s - 1 H z - 1 / 2 and $${20}\,\upmu \textrm{m}\,\textrm{s}^{-2} \textrm{H}\textrm{z}^{-1/2}$$ 20 μ m s - 2 H z - 1 / 2 respectively in its compact version and three orders of magnitude better ( $${5}\,\textrm{nrad}\, \textrm{s}^{-1} \textrm{H}\textrm{z}^{-1/2}$$ 5 nrad s - 1 H z - 1 / 2 and $${10}\,\textrm{pm}\,\textrm{s}^{-2} \textrm{H}\textrm{z}^{-1/2}$$ 10 pm s - 2 H z - 1 / 2 , respectively) with its high-performance version. Here, we compare these instrument performances with the precision required to measure rotation and tides in order to improve our knowledge of the interior of nine celestial bodies identified as targets for future space missions: Dimorphos, Phobos, Europa, Io, Titan, Enceladus, Triton, the Moon and Mars. Results indicate that Phobos, the Moon, and Mars cannot be investigated with the compact model, but that the interior of the other bodies can be constrained through measurements of rotation rate, and/or centrifugal acceleration, and/or tidal acceleration. We also find that the high-performance prototype instrument is suitable for acceleration measurements for all nine bodies, but not adequate for inferring interior constraints from rotation rate measurements for Triton, the Moon, and Mars. The signatures of the interior in the rotation rate and centrifugal and tidal accelerations also provide scientific requirements for future developments of 6DoF motion sensors for these nine bodies. Graphical Abstract

Mainly designed to study minor atmospheric species in the Martian atmosphere, the Nadir and Occultation for MArs Discovery (NOMAD) instrument suite onboard the 2016 ExoMars Trace Gas Orbiter (TGO) can also be exploited for surface ice detection. In this work, we investigate the nadir observations of the NOMAD infrared channel from the Martian Years 34 to 36 (Mars 2018 to December 2022), especially for CO2 ice detection. Based on Oliva et al. (2022), we present an updated method taking advantage of the 2.7 mu m absorption band for surface ice detection by selecting the diffraction orders 190, 169, 168 and 167. We focus the analysis on the Southern polar cap and define its boundaries during its sublimation phase in MY34-36. Globally, seasonal changes seem repeatable for MY34-36. Moreover, we show the potential of the 2.29 mu m absorption band for surface CO2 ice identification through the diffraction order 193. We define a pseudo-band depth as a good proxy for CO2 ice detection. Following a semi-qualitative approach, we attempt to reproduce such spectra by using the Planetary Spectrum Generator (PSG) model in order to estimate CO2 ice equivalent grain size. For the selected periods, the estimations are in the order of centimetres, which is in agreement with previous studies using spectral observations of OMEGA, CRISM and TES instruments.

This project takes advantage of the NOMAD spectrometer observations, on board the 2016 ExoMars Trace Gas Orbiter. These observations will help to determine the Martian surface properties. This work focuses on surface ice detection.ExoMars is an ESA-Roscosmos joint mission consisting of an orbiter (Trace Gas Orbiter - TGO). The Nadir and Occultation for Mars Discovery (NOMAD) is one of the four instruments on board TGO. The instrument is a suite of three spectrometers designed to observe the atmosphere and the surface of Mars in the UV, visible and IR. For this study, the Limb, Nadir and Occultation (LNO) channel, operating in the IR, is selected [1,3]. Thanks to the nadir geometry, the NOMAD-LNO spectrometer can provide information on Martian surface, in particular on surface ices. Ice deposits exhibit specific IR signatures in the 2.3-3.8 μm range of NOMAD-LNO, which reveals information on their composition, texture and size of the ice grains [4].We use the NOMAD nadir data to map the presence of ices for MY 34 (Ls = 150°-360°) and MY 35 (Ls = 0°-210°) with a solar zenith angle below 80 degrees. Raw data have been calibrated by using full solar scans to provide the reflectance factor which is the spectral radiance measured by NOMAD divided by the solar spectral irradiance corrected for the incident angle.We use spectral indices based on reflectance factor similar to G. Bellucci et al., 2019 [2]. An ice index is defined as a ratio between two orders taking into account the albedo variations. The objective is to remove the atmospheric effects. In this work, we propose different indices to detect the two polar caps and the sublimation process. The orders 194 (2275 - 2293.2 nm), 189 (2335.2 - 2363.9 nm) and 167 (2642.8 - 2663.9 nm) have been selected. We call Index 1, the index defined as 189/167 [5], and Index 2, 194/167. While orders 194 and 189 have wavelengths on the radiance continuum, the order 167 is located on the shoulder of 2.7 µm CO2/H2O ices absorption.All LNO observations considered in this study with the orders given above are organized as Latitude and Solar longitude maps, called seasonal maps. Figure 1 shows the seasonal map for Index 1 during MY 34. The good coverage allows observing the sublimation process for the southern polar cap between the 150°-250° solar longitude. Due to the lack of observations, the detection of the northern polar cap does not seem very clear. Nevertheless, Index 1 gives high values between Ls = 320°-340° and for latitudes above 60° which should correspond to the polar cap in that region. Figure 2 shows the same index for MY 35. The sublimation process can be observe for the northern polar cap between Ls= 0°-30°. A small detection of the southern polar cap can be see after Ls = 200°.Figure 3 shows the seasonal map for Index 2. There are observations only for MY35 and many gaps are present due to the lack of observations. Nevertheless, the two polar caps can be observe in the plot. Index 2 gives the two sublimation processes between Ls = 0°-30° for the northern polar cap and around Ls = 200° for the southern polar cap.Those new results extend the temporal and spatial of the previous work (G. Bellucci et al., 2019 [2] and L. Ruiz Lozano et al., 2019 [5]). Figure 1: Seasonal map of Index 1 for MY 34 showing the sublimation of the southern polar cap. Figure 2: Seasonal map of Index 1 for MY 35 showing the sublimation of the northern polar cap. Figure 3: Seasonal map of Index 2 for MY 35 showing the sublimation of the two polar caps. AcknowledgementsThe NOMAD experiment is led by the Royal Belgian Institute for Space Aeronomy (IASB-BIRA), assisted by Co-PI teams from Spain (IAA-CSIC), Italy (INAF-IAPS), and the United Kingdom (Open University). This project acknowledges funding by the Belgian Science Policy Office (BELSPO), with the financial and contractual coordination by the ESA Prodex Office (PEA 4000103401, 4000121493), by Spanish Ministry of Science and Innovation (MCIU) and by European funds under grants PGC2018-101836-B-I00 and ESP2017-87143-R (MINECO/FEDER), as well as by UK Space Agency through grants ST/R005761/1, ST/P001262/1, ST/R001405/1 and ST/R001405/1 and Italian Space Agency through grant 2018-2-HH.0. This work was supported by the Belgian Fonds de la Recherche Scientifique – FNRS under grant number 30442502 (ET_HOME). The IAA/CSIC team acknowledges financial support from the State Agency for Research of the Spanish MCIU through the ‘Center of Excellence Severo Ochoa’ award for the Instituto de Astrofísica de Andalucía (SEV-2017-0709). US investigators were supported by the National Aeronautics and Space Administration. Canadian investigators were supported by the Canadian Space Agency.References[1] A.C. Vandaele et al., Optical and radiometric models of the NOMAD instrument part I: the UVIS channel. Optics Express, 23(23):30028–30042,2015.[2] G. Bellucci et al., TGO/NOMAD Nadir observations during the 2018 global dust storm event, EPSC-DPS 2019[3] E. Neefs et al., NOMAD spectrometer on the ExoMars trace gas orbiter mission: part 1—design, manufacturing and testing of the infrared channels. Applied optics, 54(28):8494–8520, 2015.[4] Y. Langevin et al., Observations of the south seasonal cap of Mars during recession in 2004–2006 by the OMEGA visible/near-infrared imaging spectrometer[5] L. Ruiz Lozano, Use of NOMAD Observations (Trace Gas Orbiter) for Mars surface depositions, EPSC-DPS 2019

In this paper, we want to extend an alternative working method (Degryse and Dehant, 1993) for determining analytically core modes and frequencies of core oscillations to a non-rotating spherically symmetric Earth that might be compressible and that has a radially dependent viscosity. Since the viscosity of the outer core is considered to be small, an appropriate way to study its impact is by using boundary-layer theory (Van Dyke, 1964; Kevorkian and Cole, 1981). The representation of numerical values of eigenperiods is focused on the eigenperiod of the principal translational (Slichter, 1961) mode of free oscillation of the solid inner core. For a small value of the outer core viscosity of the order of 10−10kg/(ms), we obtained a non-rotating Slichter period of 4.886 hr, which only slightly differs from the result of the non-viscous case. For an increasing viscosity, the eigenperiods will slightly increase.