Harmonic Degree Research Articles

The tidal excitation ofrmodes in a solar-type star orbited by a giant planet companion and the effect on orbital evolution I: the aligned case

ABSTRACTIt has been suggested that tidal interaction is important for shaping the orbital configurations of close orbiting giant planets. The excitation of propagating waves and normal modes (dynamical tide) will be important for estimating time-scales for orbital evolution. We consider the tidal interaction of a Jupiter mass planet orbiting a solar-type primary. Tidal and rotational frequencies are assumed comparable making the effect of rotation important. Although centrifugal distortion is neglected, Coriolis forces are fully taken into account. We focus in detail on the potentially resonant excitation of r modes associated with spherical harmonics of degrees three and five. These are mostly sited in the radiative core but with a significant response in the convective envelope where dissipation occurs. Away from resonance significant orbital evolution over the system lifetime is unlikely. However, tidal interaction is enhanced near resonances and the orbital evolution accelerated as they are passed through. This speed up may be sustained if near resonance can be maintained. For close orbits with primaries rotating sufficiently rapidly, this could arise from angular momentum loss and stellar spin-down through a stellar wind bringing about significant orbital evolution over the system lifetime.

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).

Pulsating hydrogen-deficient white dwarfs and pre-white dwarfs observed with TESS

Context. The TESS space mission has recently demonstrated its great potential to discover new pulsating white dwarf and pre-white dwarf stars, and to detect periodicities with high precision in already known white-dwarf pulsators. Aims. We report the discovery of two new pulsating He-rich atmosphere white dwarfs (DBVs) and present a detailed asteroseismological analysis of three already known DBV stars employing observations collected by the TESS mission along with ground-based data. Methods. We processed and analyzed TESS observations of the three already known DBV stars PG 1351+489 (TIC 471015205), EC 20058−5234 (TIC 101622737), and EC 04207−4748 (TIC 153708460), and the two new DBV pulsators WDJ152738.4−50207.4 (TIC 150808542) and WD 1708−871 (TIC 451533898), whose variability is reported for the first time in this paper. We also carried out a detailed asteroseismological analysis using fully evolutionary DB white-dwarf models built considering the complete evolution of the progenitor stars. We constrained the stellar mass of three of these target stars by means of the observed period spacing, and derived a representative asteroseismological model using the individual periods, when possible. Results. We extracted frequencies from the TESS light curves of these DBV stars using a standard pre-whitening procedure to derive the potential pulsation frequencies. All the oscillation frequencies that we found are associated with g-mode pulsations with periods spanning from ∼190 s to ∼936 s. We find hints of rotation from frequency triplets in some of the targets, including the two new DBVs. For three targets, we find constant period spacings, which allowed us to infer their stellar masses and constrain the harmonic degree ℓ of the modes. We also performed period-to-period fit analyses and found an asteroseismological model for three targets, with stellar masses generally compatible with the spectroscopic masses. Obtaining seismological models allowed us to estimate the seismological distances and compare them with the precise astrometric distances measured with Gaia. We find a good agreement between the seismic and the astrometric distances for three stars (PG 1351+489, EC 20058-5234, and WD 1708-871); although, for the other two stars (EC 04207-4748 and WD J152738.4-50207), the discrepancies are substantial. Conclusions. The high-quality data from the TESS mission continue to provide important clues which can be used to help determine the internal structure of pulsating pre-white dwarf and white dwarf stars through the tools of asteroseismology.

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.

Topological Modes in Stellar Oscillations

Stellar oscillations can be of topological origin. We reveal this deep and so far hidden property of stars by establishing a novel parallel between stars and topological insulators. We construct an Hermitian problem to derive the expression of the stellar acoustic–buoyant frequency S of nonradial adiabatic pulsations. A topological analysis then connects the changes of sign of the acoustic–buoyant frequency to the existence of Lamb-like waves within the star. These topological modes cross the frequency gap and behave as gravity modes at low harmonic degree ℓ and as pressure modes at high ℓ. S is found to change sign at least once in the bulk of most stellar objects, making topological modes ubiquitous across the Hertzsprung–Russell diagram. Some topological modes are also expected to be trapped in regions where the internal structure varies strongly locally.

Tidal Constraints on the Martian Interior

AbstractWe compare several recent Martian interior models and evaluate how these are impacted by the tidal constraints provided by the Love number k2 and the secular acceleration in longitude s of its main moon, Phobos. The expression of the latter is developed up to harmonic degree 5 to match the accuracy of the current observations. We match a number of current interior structure models to the recent measurements of the tidal parameters and derive estimations of the possible core radius, temperature profile, and attenuation in the Martian interior. Our estimation of the core radius is 1,820 ± 80 km, consistent with recent seismic measurements. The attenuation profiles in the Martian interior at the main tidal period of Phobos are similar between the considered models, giving a range for the degree‐2 bulk tidal attenuation Q2 = 93.0 ± 8.40 but diverge at seismic frequencies. At seismic frequencies, model shear attenuation Qμ ranges between 100 and 4,000 in the lower mantle, so that a measurement of seismic shear attenuation could be used as an effective means for distinguishing between the models considered. Other constraints such as elastic lithosphere thickness and Chandler Wobble period favor a thicker elastic lithosphere and models with a frequency dependence α of the shear attenuation between 0.15 and 0.4. Improved constraints on the Martian interior should be possible with additional seismic and radio observations from the InSight mission.

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).

Preference of harmonic loudness degree on piano timbre.

This study exploratively conducted two investigations of timbre influenced by harmonic loudness. Investigation 1 examined piano timbre preference between non-processed melodies and two kinds of adjustments created via an equalizer–a basic tool used for sound engineering and for producing audio materials. Using the paired comparison method, 98 respondents were surveyed. The results show that in the bass range of the piano sound, the preferred audio material was that produced via an equalizer by reducing the loudness of the 7th and 9th harmonics by 15 dB, while enhancing the loudness of the 8th harmonic by 15 dB. Investigation 2 examined three degrees of the processed melodies—adjusting the harmonics by 15, 10, and 5 dB. The results show that the 15 dB change was the best, while the 10 dB change was the worst. These results provide a new approach to improving the sound timbre for sound engineering and artificial intelligence music production.

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.

Crustal anisotropy beneath southeastern Tibet inferred from directional dependence of receiver functions

The present study reveals depth-dependent crustal anisotropic signatures beneath southeastern Tibet. It is used to characterize the rheology of major faults and shear zones, which is important for understanding crustal deformation and geodynamic processes beneath the study area. The depth-dependent anisotropic orientations have been investigated based on the directional dependence of radial and tangential receiver functions (RFs). To achieve our objective, we first computed 3683 good-quality P-RFs from 174 teleseismic earthquakes (M ≥ 5.5) recorded within epicentral distance range of 30∘ to 90∘ at 70 seismic stations of the Eastern Syntaxis experiment (XE Network). After that, we employed the harmonic decomposition technique at each seismic station to retrieve the first (k = 0), second (k = 1) and third (k = 2) degree harmonics from the RF dataset. Our study characterizes the type (horizontal or plunging) of the symmetry axis. The anisotropic axes of the upper crust (0–20 km) appear to vary from approximately N-S to NE-SW. They are usually orthogonal to orientations of major faults and suture zones in the region, implying the effect of structure-induced anisotropy. It can be explained by regularly oriented cracks or macroscopic structure alignment along the major faults. The anisotropic orientations of the middle crust (20–40 km) are NE-SW to E-W direction, reflecting a different pattern than those estimated in the upper crust. The lower crustal (40–70 km) anisotropic pattern (E-W or ESE-WNW direction) exhibits distinct orientations than the upper and middle crust. The crystal preferred orientations (CPO) of the mica and amphibole minerals are likely to cause anisotropy observed at mid-to-lower crustal depth ranges, emphasizing the role of ductile deformation due to material movement towards the east underneath southeastern Tibet. Our results, along with S(K)KS and direct S-waves splitting signatures, suggest mid-to-lower crust and lithospheric mantle material movement towards the east, while the discrepancies in anisotropic symmetric axes directions may be indicative for the partial coupling between the crust and upper mantle beneath the region.

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.

Effects of Intense Laser Field on Electronic and Optical Properties of Harmonic and Variable Degree Anharmonic Oscillators

In this paper, we calculated the electronic and optical properties of the harmonic oscillator and single and double anharmonic oscillators, including higher-order anharmonic terms such as the quartic and sextic under the non-resonant intense laser field. Calculations are made within the effective mass and parabolic band approximations. We have used the diagonalization method by choosing a wave function based on the trigonometric orthonormal functions to find eigenvalues and eigenfunctions of the electron confined within the harmonic and anharmonic oscillator potentials under the non-resonant intense laser field. A two-level approach in the density matrix expansion is used to calculate the linear and third-order nonlinear optical absorption coefficients. Our results show that the electronic and optical properties of the structures we focus on can be adjusted to obtain a suitable response to specific studies or aims by changing the structural parameters such as width, depth, coupling between the wells, and applied field intensity.

MMC Admittance Model Simplification Based on Signal-Flow Graph

Although the impedance/admittance model of modular multilevel converter (MMC) considering internal harmonic is discussed extensively in the literature, few attentions have been focused on how to determine the number of harmonics to be included in the model. This article fills this gap by developing the MMC signal-flow graph (SFG), providing the analytical transfer function to quantitatively estimate the attenuation degree of MMC harmonics, so as to guide the model truncation. Besides, SFG offers an intuitive understanding on the mathematical characteristic of MMC, including the conjugate symmetry and the effect of phase reference on admittance, which helps to reduce the modeling calculation. Moreover, conversion formula of MMC multiple input multiple output and single input single output admittance is derived based on the truncated model to further simplify the stability analysis. The correctness of this article has been verified by simulation and experiment.

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.

A multi-instrument investigation of the frequency stability of oscillations above the acoustic cut-off frequency with solar activity

ABSTRACT Below the acoustic cut-off frequency, oscillations are trapped within the solar interior and become resonant. However, signatures of oscillations persist above the acoustic cut-off frequency, and these travelling waves are known as pseudo-modes. Acoustic oscillation frequencies are known to be correlated with the solar cycle, but the pseudo-mode frequencies are predicted to vary in antiphase. We have studied the variation in pseudo-mode frequencies with time systematically through the solar cycle. We analysed Sun-as-a-star data from Variability of Solar Irradiance and Gravity Oscillations (VIRGO), and Global Oscillations at Low Frequencies (GOLF), as well as the decomposed data from Global Oscillation Network (GONG) for harmonic degrees 0 ≤ l ≤ 200. The data cover over two solar cycles (1996–2021, depending on instrument). We split them into overlapping 100-d long segments and focused on two frequency ranges, namely 5600–$6800\, \rm \mu Hz$ and 5600–$7800\, \rm \mu Hz$. The frequency shifts between segments were then obtained by fitting the cross-correlation function between the segments’ periodograms. For VIRGO and GOLF, we found no significant variation of pseudo-mode frequencies with solar activity. However, in agreement with previous studies, we found that the pseudo-mode frequency variations are in antiphase with the solar cycle for GONG data. Furthermore, the pseudo-mode frequency shifts showed a double-peak feature at their maximum, which corresponds to solar activity minimum, and is not seen in solar activity proxies. An, as yet unexplained, pseudo-periodicity in the amplitude of the variation with harmonic degree l is also observed in the GONG data.