Spherical Harmonic Degree Research Articles

Designing Satellite Constellations to Observe Earth’s Time-Varying Gravity from Clock Frequency Comparisons

The nondominated sorting genetic algorithm III (NSGA-III) is used to design nonsymmetric satellite constellations to observe large-scale high-frequency gravity variations that would serve as complementary systems to other dedicated gravity field mapping missions such as the Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow-On missions. The functionality of the NSGA-III method is validated by reproducing optimal surface-coverage satellite constellations from the literature. Then, NSGA-III is used to generate nondominated constellations tuned to detect short-term long-wavelength gravity field variations. The constellations are used to generate subweekly gravity field solutions from clock frequency comparison measurements. The effectiveness of NSGA-III for optimizing nonsymmetric constellations was demonstrated because the constellations obtained from NSGA-III were consistently superior in performance than traditional symmetric constellations. Furthermore, the results showed that the time-varying signals of interest (that is, gravity coefficients through the spherical harmonic degree and order of seven at daily intervals) can be unambiguously resolved by assuming an optical clock with a fractional frequency uncertainty representative of current technology. The ability to monitor these short-term long-wavelength signals on a global scale would observe a spectrum of the gravity field inherently undetectable by GRACE-like missions, and it would provide new insight to the understanding of global mass redistribution processes.

Small satellite formations and constellations for observing sub-daily mass changes in the Earth system

SUMMARY Satellite gravity missions so far are medium size satellites consisting of one or a pair of satellites flying in near-polar or sun-synchronous orbital planes. Due to the limited observation geometry and the related space–time sampling, high-frequency non-tidal mass variation signals from atmosphere and ocean cannot be observed and cause temporal aliasing. For current single-pair satellite gravimetry missions as GRACE and GRACE Follow-On (GRACE-FO) temporal aliasing is the limiting factor and represents the major error source in the gravity field time-series. Adding a second inclined satellite pair to a GRACE-like polar pair (Bender constellation) currently is the most promising solution to increase the spatio-temporal resolution and to significantly reduce the temporal aliasing error. This shall be implemented with the MAGIC mission in future. With the ongoing developments in miniaturization of satellites and gravity-relevant instruments (accelerometers and intersatellite ranging), in future constellations of multiple small satellite pairs may solve this problem even beyond the capabilities of a Bender constellation. Therefore, in this study the capabilities of such constellations flying in specific formations are investigated in order to enable a retrieval of the temporal gravity field on short time scales. We assess the performance of up to 18 satellite pairs. The satellite configurations cover satellite pairs in polar and inclined orbits flying in pair-wise or pearl-string formation with varying mean anomalies and right ascensions of the ascending node (RAAN). As future potential miniaturized instruments optomechanical accelerometers with similar performance as those flying on GRACE-FO are a candidate, while for the intersatellite ranging instrument still some technological development is required. Therefore, in this study a microwave ranging system equivalent to the GRACE and GRACE-FO instruments performance is taken as baseline assuming that such instruments can be miniaturized in future as well. In numerical closed-loop simulations, up to nine different satellite configurations with up to 18 satellite pairs are evaluated based on the retrieval of the non-tidal temporal gravity field on a monthly basis. From the simulation results it is concluded that the best-performing satellite constellation of 18 polar satellite pairs already is outperformed by a typical Bender-like constellation of 1 polar and 1 inclined pair. In general, we identify that increasing the number of satellite pairs leads to an improved gravity field retrieval, either at low spherical harmonic degree and order (d/o) by the shift in RAAN or at high d/o by the shift in mean anomaly. By a two-step simulation approach, co-estimating also (sub-)daily gravity fields for selected configurations with a large number of satellite pairs distributed equally over the globe, it is possible to retrieve stand-alone gravity fields at 24, 12 and 6 hr temporal resolution. Ultimately it is concluded that a network of miniaturized satellites with instrument performances similar to GRACE-FO and flying in a well-defined constellation has the potential to observe (sub-)daily mass variations and therefore could drastically reduce the problem of temporal aliasing due to high frequency mass variations in the Earth system.

Gravity degree–depth relationship using point mass spherical harmonics

SUMMARYRelationships between the degree of a spherical harmonic model of the gravitational field of a body and the depth of a source expressed as a density contrast can be used to study the structure of features. Here, we show that the gravitational acceleration per spherical harmonic degree of a constant density source has an extremum that depends on the depth of the source. Using the spherical harmonics expansion for a point mass source, we use this to derive a degree–depth relationship. Our relationship resembles an earlier one derived by Bowin, with substantial differences at the lower degrees. We also find that a recent relationship derived by Deng et al. overestimates the source depth. The relationship that we derive relates spherical harmonic degree n to depth d for a planet of radius R according to $d = (1-e^{\frac{-1}{n+1}})R$, which simplifies to d = R/(n + 1) for high degrees. We support our new relationship with synthetic models of a density contrast in a planet. We also show how the differences between our relationship and that of Bowin affect band-filtered gravity, for example when inspecting the upper 100 km of the Moon. Using point masses in our modelling results in an approximate relationship where in reality sources can be deeper than estimated, since any source contributes to all spherical harmonic degrees. The use of the contribution per individual degree however provides an intuitive relationship between spherical harmonic degree and depth that can be used to place relative bounds on source depths or to determine the bounds on spherical harmonic expansions when band-filtering gravity field models.

Simulation for MSS-2 low-perigee elliptical orbit satellites: an example of lithospheric magnetic field modelling

A future constellation of at least four geomagnetic satellites (designated Macau Scientific Satellite-1 (MSS-1) and Macau Scientific Satellite-2 (MSS-2)) was recently proposed, to continue high-quality geomagnetic observations in the post-Swarm period, focusing especially on collecting data that will provide a global, three-dimensional survey of the geomagnetic field. In this paper, we present a simulation of two years of orbits (2020.01.01−2022.01.01) of two satellites (tentatively denoted as MSS-2) that are constellated in elliptical (200 × 5,300 km) low-perigee orbits. By comparing error variances of Gauss coefficients, we investigate the sensitivity of lithospheric magnetic field modelling to data collected from various satellite orbits, including a near circular reference orbit of 300 × 350 km, and elliptical orbit of 180 × 5,300 km, 220 × 5,300 km, 200 × 3,000 km and 200 × 1,500 km. We find that in two years the two MSS-2 satellites can collect 35,000 observations at altitude below 250 km, data that will be useful in advancing the quality of lithospheric magnetic field modelling; this number of observations reflects the fact that only 4.5% of the flight time of these satellites will be below 250 km (just 6.4% of their flight time below 300 km). By combining observations from the MSS-2 satellites’ elliptical orbits of 200 × 5,300 km with observations from a circular reference orbit, the variance of the geomagnetic model can be reduced by a factor of 285 at spherical harmonic degree <italic>n</italic> = 200 and by a factor of 1,300 at <italic>n</italic> = 250. The planned lower perigee of their orbits allows the new satellites to collect data at unprecedentedly lower altitudes, thus dramatically improving the spatial resolution of satellite-derived lithospheric field models, (up to 80% at <italic>n</italic>=150). In addition, lowering the apogee increases the time interval during which the satellites fly at near-Earth altitudes, thus improving the model predictions at all spherical harmonic degrees (around 52%−62% at <italic>n</italic> = 150). The upper limit of the expected improvement to the field model at the orbital apogee is not as good as at the perigee. However, data from the MSS-1 orbit can help fill the gap between data from the MSS-2 orbits and from the circular reference orbit for the low-degree part of the model. The feasibility of even lower-altitude flight requires further discussion with satellite engineers.

The effect of model errors in ensemble sequential assimilation of geomagnetic field

Earth’s magnetic field is generated in the fluid outer core through the dynamo process. Over the last decade, data assimilation has been used to retrieve the core dynamics and predict the evolution of the geomagnetic field. The presence of model errors in the geomagnetic data assimilation is inevitable because current numerical geodynamo models are still far from realistic core dynamics. In this paper, we investigate the effect of model errors in geomagnetic data assimilation based on ensemble Kalman filter (EnKF). We construct two dynamo models with different control parameters but exhibiting similar force balance and magnetic morphology at the core surface. We then use one dynamo model to generate synthetic observations and the other as the forward model in EnKF. Our test experiments show that the EnKF approach with the pre-setting model errors can nevertheless recover large-scale core surface flow and make a rough short-term (5-year) prediction. However, the data assimilation in the presence of model errors cannot keep improving the core state even though new observations are available. Motivated by the planned Macau Science Satellite-1, which is expected to provide improved internal geomagnetic field model, we also perform a test experiment using synthetic observations up to spherical harmonic degree <inline-formula><tex-math id="M1">\begin{document}$\ell=18$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-lijinfeng_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-lijinfeng_M1.png"/></alternatives></inline-formula>. Our results indicate that high-resolution observations are crucial in reconstructing small scale flow.

Assessment of gravity field recovery from a quantum satellite mission with atomic clocks and cold atom gradiometers

The aim of the MOCAST+ (MOnitoring mass variations by Cold Atom Sensors and Time measures) project, which was carried out during the years 2020–2022, was the investigation of the performance of a gravity field mission based on the integration of atomic clocks and cold atom interferometers. The idea was that the combined observations of the two sensors would be beneficial for the detection and monitoring of geophysical phenomena which have an impact on the time-variable part of the Earth gravity field models. Several different mission scenarios were simulated, considering different satellite configurations such as a Gravity Recovery and Climate Experiment (GRACE)-class formation and a Bender-class formation with either two or three in-line satellites along each orbit. Moreover, different atomic species (rubidium and strontium), different inter-satellite distances, different noise power spectral densities, and different observation rates were taken into account. For the gravity field estimation from the simulated data, the space-wise approach was exploited. The results showed that, as it could be expected, the Bender configuration provides significantly better monthly gravity field solutions, as compared to a ‘nominal’ configuration with two or three satellites in a GRACE-class formation. In this way, and pushing the quantum sensors technology to its limits, it is in fact possible to obtain results which are comparable with those from GRACE at low harmonic degrees, and are better at higher degrees with positive effects in the detectability of localized time variable phenomena, as well as in the determination of the static gravity field at a higher maximum spherical harmonic degree than the one achieved by Gravity Field and Steady-State Ocean Circulation Explorer (of course considering an equivalent mission life-time).

Spherical harmonic decomposition and interpretation of the shapes of the small Saturnian inner moons

Context. The Cassini-Huygens space mission made a series of observations of Saturn’s small satellites during its grand finale stage. These measurements were performed in order to study the shape, geology, and surface composition of the small satellites as well as to study the impact of the environment, in particular the rings, on these small bodies. Aims. The purpose of this study is to focus on the shape analysis of the small Saturnian satellites in order to describe their global figure and large-scale topography, as well as to deduce fundamental quantities, gravity field, and amplitude of the diurnal libration by assuming that the bodies are homogeneous. Methods. We used two approaches in this study. On the one hand, we directly exploited the Cassini images of the small satellites by performing limb measurements and deducing a confidence interval on the shape measurements. On the other hand, we used previously published shape models which combine limb measurements and control points. These shape models were then decomposed and described in spherical harmonics. Results. We found that the shape of the small satellites can be described with a confidence interval between 50 and 150 m. The low degree in spherical harmonics (degree 2) indicated that Telesto, Pandora, Pan, Janus, and Helene have a degree 2 shape close to the Omega sequence, which was defined recently, where the potential is constant along a meridian perpendicular to the longest axis. The degree 2 shape of Epimetheus, on the other hand, is close to the Roche sequence. In contrast, Prometheus, Calypso, and Atlas are in the Low-Brown region. The root mean square spectrum and spherical harmonic maps then allowed us to describe the topography of the satellites, and in particular to highlight equatorial ridges for some satellites including Daphnis. Finally, our estimates of the libration amplitude in the homogeneous case provide values in agreement with previously published librational measurements for Epimetheus while highlighting the proximity of the resonance for Epimetheus, Pandora, and Prometheus. Conclusions. The high resolution images of the internal satellites have allowed us to describe the geology and the geophysics of these bodies. Future comparison of these amplitudes with new librational measurements deduced, for example, by the astrometric method, will allow us to obtain information on the internal structure of these bodies. Similar studies could be carried out on the internal satellites of Jupiter using images from the Europa Clipper (NASA) or JUICE (ESA) missions.

Modelling Earth’s lithospheric magnetic field using satellites in low-perigee elliptical orbits

SUMMARYThe sensitivity of magnetic measurements taken by satellites in elliptical orbits to the lithospheric magnetic field is studied by comparing the formal error variances of the lithospheric Gauss coefficients for various satellite orbital constellations. Analytical expressions are presented for the variances of the Gauss coefficients when either all three magnetic vector components or only the radial component are used. We compare the results obtained using a satellite in a near-polar circular orbit at 350 km altitude with those from a satellite in an elliptical orbit with perigee at 140 km (and apogee at 1500 km) and find that the latter leads to Gauss coefficient variances at spherical harmonic degree n = 180 (corresponding to a horizontal wavelength of λ = 220 km) that are 104 times smaller compared to those derived from a similar number of data measured at 350 km altitude. The improvements in variance ratio at degree n = 145 (λ = 275 km) and n = 110 (λ = 360 km) are 103 and 102, respectively. These findings are supported by an analysis of synthetic magnetic data along simulated satellite orbits from which the lithospheric Gauss coefficients are estimated and compared with the original ones used to generate the synthetic data. Coefficients at degree n are successfully determined if the power of the difference between retrieved and original coefficients at that degree is smaller than the power of the lithospheric field (i.e. of the input coefficients). Using 3 yr of simulated data we conclude that magnetic measurements from a satellite in an elliptical orbit with perigee at 140 km allow for a reliable determination of the lithospheric field up to spherical harmonic n ≈ 170 while a satellite in a circular orbit at 350 km height only enables lithospheric field modelling up to n ≈ 100. The analysis demonstrates that low-altitude magnetic data collected by satellites in low-perigee elliptical orbits—although only available for a fraction of each orbit—enable improved global lithospheric field modelling at spatial wavelengths well beyond what is currently possible with data from satellites in circular orbits that do not reach such low altitudes. We applied the approach to the orbital configuration proposed for the Daedalus satellite mission (140 km perigee); the method will however also help in the preparation for other satellite missions in near-polar low-perigee elliptical orbits like the Macau Science Satellite pair MSS-2A and MSS-2B (perigee of 200 km or lower).

Kalmag: a high spatio-temporal model of the geomagnetic field

We present the extension of the Kalmag model, proposed as a candidate for IGRF-13, to the twentieth century. The dataset serving its derivation has been complemented by new measurements coming from satellites, ground-based observatories and land, marine and airborne surveys. As its predecessor, this version is derived from a combination of a Kalman filter and a smoothing algorithm, providing mean models and associated uncertainties. These quantities permit a precise estimation of locations where mean solutions can be considered as reliable or not. The temporal resolution of the core field and the secular variation was set to 0.1 year over the 122 years the model is spanning. Nevertheless, it can be shown through ensembles a posteriori sampled, that this resolution can be effectively achieved only by a limited amount of spatial scales and during certain time periods. Unsurprisingly, highest accuracy in both space and time of the core field and the secular variation is achieved during the CHAMP and Swarm era. In this version of Kalmag, a particular effort was made for resolving the small-scale lithospheric field. Under specific statistical assumptions, the latter was modeled up to spherical harmonic degree and order 1000, and signal from both satellite and survey measurements contributed to its development. External and induced fields were jointly estimated with the rest of the model. We show that their large scales could be accurately extracted from direct measurements whenever the latter exhibit a sufficiently high temporal coverage. Temporally resolving these fields down to 3 hours during the CHAMP and Swarm missions, gave us access to the link between induced and magnetospheric fields. In particular, the period dependence of the driving signal on the induced one could be directly observed. The model is available through various physical and statistical quantities on a dedicated website at https://ionocovar.agnld.uni-potsdam.de/Kalmag/.Graphical abstract

The Internal Structure and Dynamics of Jupiter Unveiled by a High‐Resolution Magnetic Field and Secular Variation Model

AbstractUnique information about the dynamo process acting at Jupiter can be inferred by modeling and interpreting its magnetic field. Using the fluxgate magnetometer measurements acquired during the 4 years of the Juno mission, we derive a magnetic field model which describes simultaneously the main field and the secular variation (SV) up to spherical harmonic degrees 16 and 8, respectively. Apart from the Earth's, this is the first time another planetary magnetic field along with its time variation is described to such a high degree. We use properties of the power spectrum of the static field to infer the upper boundary of the dynamo convective region at 0.830 ± 0.022 Jupiter radius. The SV and correlation times are relatively comparable to the Earth's and indicate that the field is dominated by advection. The field and SV morphologies suggest zonal as well as non‐zonal deep fluid motions.

Impacts of zonal winds on planetary oscillations and Saturn ring seismology

ABSTRACT The excitation of density and bending waves in Saturn’s C ring by planetary oscillation modes presents a unique opportunity to learn about gas giant interiors and rotation. However, theoretical complications related to Saturn’s rapid and differential rotation pose a barrier to the full utilization of ring wave detections. We calculate oscillation modes using a complete, non-perturbative treatment of differential rotation modelled after Saturn’s zonal winds in self-consistently computed, polytropic equilibria. We find that previous, approximate treatments of the effects of differential rotation in Saturn overestimate shifts in the frequencies of fundamental modes (f modes) thought to be responsible for the majority of the waves detected in the C ring, due to an omitted modification of the equilibrium shape and structure of the planet by differential rotation. The bias introduced by these frequency overestimates is small, but significant relative to the uncertainties afforded by Cassini data. We additionally consider the non-perturbative effects of Saturn-like differential rotation on the rotational mixing of f modes and internal gravity modes (g modes), which is relevant to detections of multiple density waves with very closely split pattern speeds. We find that higher-order rotational effects can produce orders-of-magnitude enhancements in the surface gravitational perturbations of g modes dominated by large spherical harmonic degrees ℓ, regardless of frequency separation from the sectoral f mode. Despite this enhancement, we find that the observed fine splitting of density waves is unlikely to involve g modes dominated by ℓ ≳ 10. This restriction may aid in the inference of possible internal structures for Saturn.

Physical limits to human brain B0 shimming with spherical harmonics, engineering implications thereof.

As the MRI main magnetic field rises for improved signal-to-noise ratio, susceptibility-induced B0-inhomogeneity increases proportionally, aggravating related artifacts. Considering only susceptibility disparities between air and biological tissue, we explore the topological conditions for which perfect shimming could be performed in a Region of Interest (ROI) such as the human brain or part thereof. After theoretical considerations for perfect shimming, spherical harmonic (SH) shimming simulations of very high degree are performed, based on a 100-subject database of 1.7-mm-resolved brain fieldmaps acquired at 3T . In addition to the whole brain, shimmed ROIs include slabs targeting the prefrontal cortex, both or single temporal lobes, or spheres in the frontal brain above the nasal sinus. We show "perfect" SH shimming is possible only if the ROI can be contained in a sphere that does not enclose sources of magnetic field inhomogeneity, which are gathered at the air-tissue interface. We establish a [Formula: see text]Hz inhomogeneity hard shim limit at 7T for whole brain SH shimming, that can only be attained at shimming degree higher than 90. On the other hand, under limited power and SH degree resources, 3D region-specific shimming is shown to greatly improve homogeneity in critical zones such as the prefrontal cortex and around ear canals.

Drag and Attitude Control for the Next Generation Gravity Mission

The Next Generation Gravity Mission (NGGM), currently in a feasibility study phase as a candidate Mission of Opportunity for ESA-NASA cooperation in the frame of the Mass Change and Geo-Sciences International Constellation (MAGIC), is designed to monitor mass transport in the Earth system by its variable gravity signature with increased spatial and temporal resolution. The NGGM will be composed by a constellation of two pairs of satellites, each providing the measurement of two quantities from which the map of Earth’s gravity field will be obtained: the variation of the distance between two satellites of each pair, measured by a laser interferometer with nanometer precision; and the relative non-gravitational acceleration between the centers of mass of each satellite pair, measured by ultra-sensitive accelerometers. This article highlights the importance of the second “observable” in the reconstruction of the lower harmonics of Earth’s gravity field, by highlighting the tight control requirements in linear and angular accelerations and angular rates, and the expectable performances from the drag-free, attitude, and orbit control system (DFAOCS) obtained through an end-to-end (E2E) simulator. The errors resulting from different mission scenarios with varying levels of drag-free control and pointing accuracy are then presented, demonstrating that a high-performance accelerometer alone is not sufficient to achieve the measurement quality necessary to achieve the mission objectives, if the spacecraft does not provide to this sensor a suitable drag-free environment and a precise and stable pointing. The consequences of these different mission scenarios on the gravity field retrieval accuracy, especially for the lower spherical harmonic degrees, are computed in order to quantitatively justify the rationale for these capabilities on the NGGM spacecraft.

Degree‐Depth Relation for Planetary Gravity Field Model Based on Wavelength

AbstractIn this paper we investigate the relation between the spherical harmonic (SH) degree of gravity field model and the corresponding depth of the source body. By investigating the gravity of a buried point mass, we find that a relationship can be found between the horizontal extent of the signal (which we assume to be indicative of the wavelength) and the depth of the source. We introduce a threshold to define the gravity anomaly wavelength of an isolated mass body. The region where the gravity anomaly of an isolated mass body is larger than the threshold defines its gravity anomaly wavelength. For an isolated point mass with fixed total mass, the wavelength is a function of the depth. The ratio between the maximum wavelength and the depth at which the maximum wavelength occurs (effective resolvable depth) is invariant provided that the mass and threshold are determined. When this maximum wavelength is larger than the minimum resolvable wavelength of n‐degree gravity field, this isolated mass body is considered as a reliable density feature. Combining the relation between maximum wavelength and effective resolvable depth of the isolated mass body and the relation between minimum resolvable wavelength and SH degree of gravity field, we derived the degree‐depth relation under the limiting resolvable condition: . This relation was validated with Gravity Recovery and Interior Laboratory gravity field data at lunar Orientale basin region, and provided insights for the planetary gravity field analysis.

Spectral consistency of satellite and airborne data: Application of an equivalent dipole layer for combining satellite and aeromagnetic data sets

We present a novel approach for the combination of aeromagnetic and satellite data applying an equivalent dipole layer and spherical harmonic (SH) expansion of the dipoles. The method involves two steps: (1) inversion for the magnetic parameters of the equivalent dipole layer and (2) conversion of the magnetic parameters into SH coefficients. With this approach, SH analysis can be used for a regional study area, and, for example, the long-wavelength range of aeromagnetic data can be replaced with satellite data. We test our approach on the third, fourth, and fifth editions of the Magnetic Anomaly Map of Australia, which were releveled using independent aeromagnetic data sets for long-wavelength correction. The results indicate that the magnetic compilations leveled to long-haul control lines have a good agreement with the LCS-1 satellite model in the range of SH degree 40 to 110 (corresponding to half-wavelengths between 180 to 500 km), whereas the third edition of the Magnetic Anomaly Map of Australia demonstrates poor control over long- wavelengths in this spectral range. Our analysis suggests that even the carefully processed fifth edition would benefit from the replacement of long-wavelength data with satellite data.

Do lower mantle slabs contribute in generating the Indian Ocean geoid low?

The Indian Ocean geoid low (IOGL) is one of the lowest geoid anomalies on Earth. Several theories have been proposed to explain this geoid low, including past subduction (Nerlich et al., 2016; Rao and Kumar, 2014), subduction coupled with low velocity anomalies in the upper mantle (Spasojevic et al., 2010) and hot, low density anomaly in the upper to mid mantle depths (Reiss et al., 2017; Ghosh et al., 2017). It was also argued by Ghosh et al. (2017) that subducted slabs in the lower mantle have minimal role to play in contributing to this geoid signal. We further investigate that claim here by looking at the geoid contributed from various processes (density vs dynamic topography) as well as by inspecting the contribution from different spherical harmonic degrees. Our current findings substantiate the earlier claim that lower mantle slabs indeed play a minimal role in creating this anomalous geoid low.

A Spherical Harmonic Model of Earth's Lithospheric Magnetic Field up to Degree 1050

AbstractDetailed mapping of the Earth's magnetic field brings key constraints on the composition, dynamics, and history of the crust. Satellite and near‐surface measurements detect different length scales and are complementary. Here, we build a model, selecting and processing magnetic field measurements from the German CHAMP and ESA Swarm satellites, which we merge with near‐surface scalar anomaly data. We follow a regional approach for modeling the magnetic field measurements and next transform the series of regional models into a unique set of spherical harmonic (SH) Gauss coefficients. This produces the first global model to SH degree 1050 derived by inversion of all available measurements. The new model agrees with previous satellite‐based models at large wavelengths and fits the CHAMP and Swarm satellite data down to expected noise levels. Further assessment in the geographical and spectral domains shows the model to be stable when downward continued to the Earth's surface.

Constraining Saturn’s Interior with Ring Seismology: Effects of Differential Rotation and Stable Stratification

Abstract Normal mode oscillations in Saturn excite density and bending waves in the C ring, providing a valuable window into the planet’s interior. Saturn’s fundamental modes (f-modes) excite the majority of the observed waves, while gravito-inertial modes (rotationally modified g-modes) associated with stable stratification in the deep interior provide a compelling explanation for additional density waves with low azimuthal wavenumbers m. However, multiplets of density waves with nearly degenerate frequencies, including an m = 3 triplet, still lack a definitive explanation. We investigate the effects of rapid and differential rotation on Saturn’s oscillations, calculating normal modes for independently constrained interior models. We use a non-perturbative treatment of rotation that captures the full effects of the Coriolis and centrifugal forces, and consequently the mixing of sectoral f-modes with g-modes characterized by very different spherical harmonic degrees. Realistic profiles for differential rotation associated with Saturn’s zonal winds can enhance these mode interactions, producing detectable oscillations with frequencies separated by less than 1%. Our calculations demonstrate that a three-mode interaction involving an f-mode and two g-modes can feasibly explain the finely split m = 3 triplet, although the fine-tuning required to produce such an interaction generally worsens agreement with seismological constraints provided by m = 2 density waves. Our calculations additionally demonstrate that sectoral f-mode frequencies are measurably sensitive to differential rotation in Saturn’s convective envelope. Finally, we find that including realistic equatorial antisymmetry in Saturn’s differential rotation profile couples modes with even and odd equatorial parity, producing oscillations that could in principle excite both density and bending waves simultaneously.

Time-Variable Gravity Field from the Combination of HLSST and SLR

The Earth’s time-variable gravity field is of great significance to study mass change within the Earth’s system. Since 2002, the NASA-DLR Gravity Recovery and Climate Experiment (GRACE) and its successor GRACE follow-on mission provide observations of monthly changes in the Earth gravity field with unprecedented accuracy and resolution by employing low-low satellite-to-satellite tracking (LLSST) measurements. In addition to LLSST, monthly gravity field models can be acquired from satellite laser ranging (SLR) and high-low satellite-to-satellite tracking (HLSST). The monthly gravity field solutions HLSST+SLR were derived by combining HLSST observations of low earth orbiting (LEO) satellites with SLR observations of geodetic satellites. Bandpass filtering was applied to the harmonic coefficients of HLSST+SLR solutions to reduce noise. In this study, we analyzed the performance of the monthly HLSST+SLR solutions in the spectral and spatial domains. The results show that: (1) the accuracies of HLSST+SLR solutions are comparable to those from GRACE for coefficients below degree 10, and significantly improved compared to those of SLR-only and HLSST-only solutions; (2) the effective spatial resolution could reach 1000 km, corresponding to the spherical harmonic coefficient degree 20, which is higher than that of the HLSST-only solutions. Compared with the GRACE solutions, the global mass redistribution features and magnitudes can be well identified from HLSST+SLR solutions at the spatial resolution of 1000 km, although with much noise. In the applications of regional mass recovery, the seasonal variations over the Amazon Basin and the long-term trend over Greenland derived from HLSST+SLR solutions truncated to degree 20 agree well with those from GRACE solutions without truncation, and the RMS of mass variations is 282 Gt over the Amazon Basin and 192 Gt in Greenland. We conclude that HLSST+SLR can be an alternative option to estimate temporal changes in the Earth gravity field, although with far less spatial resolution and lower accuracy than that offered by GRACE. This approach can monitor the large-scale mass transport during the data gaps between the GRACE and the GRACE follow-on missions.

Improved Determination of Europa's Long‐Wavelength Topography Using Stellar Occultations

AbstractEuropa Clipper will arrive at Jupiter at the end of this decade and will explore Europa through a series of flybys. One of its many goals is to characterize Europa's topography and global shape using the Europa Imaging System and Radar for Europa Assessment and Sounding: Ocean to Near‐surface (REASON) instruments. In addition, Europa Clipper's UV Spectrograph will observe stars pass behind (be occulted by) Europa. The spectrograph has sufficiently precise timing, corresponding to a topographic precision of order meters, that these occultations can also serve as altimetric measurements. Because of gaps in the REASON radar altimeter coverage imposed by the flyby geometries, the addition of ∼100 occultations results in a substantial improvement in the recovery of Europa's long‐wavelength shape. Typically, five extra spherical harmonic degrees of topography can be recovered by combining occultations with radar altimetry.