Love Number H2 Research Articles

Combination of altimetry crossovers and Doppler observables for orbit determination and geodetic parameter recovery: Application to Callisto

An accurate knowledge of the orientation, the tidal deformability, and the gravity field of a celestial body is fundamental to provide constraints on its internal structure. These quantities may be retrieved by processing radiometric tracking and altimetry data from a probe in orbit around such body. This paper presents a method to combine altimetry crossovers with two-way Doppler tracking observations at normal equation level, using the Bernese GNSS Software and the pyXover software library. This method was applied to a proposed 200km altitude orbiter around Callisto, a privileged destination for the upcoming phase of Solar System exploration. Enhancing “standard” Doppler tracking with altimetry generally benefited both orbit determination and a joint estimation of the orientation of the north pole and of planetary librations. The retrieval of low-degree gravity field parameters was also improved by the addition of altimetry data. However, the improvements on the estimated parameters were highly dependent on the characteristics of the simulation, e.g., the underlying topography roughness. Overall, combining radioscience with altimetry data accounted for a visible reduction of correlations among estimated parameters, while also allowing for a consistent estimation of the “vertical” Love number h2 along with gravity.

Deriving Mercury Geodetic Parameters With Altimetric Crossovers From the Mercury Laser Altimeter (MLA)

AbstractBased on the previous applications of laser altimetry to planetary geodesy at GSFC, we use the recently developed PyXover software package to analyze altimetric crossovers from the Mercury Laser Altimeter (MLA). Using PyXover, we place new constraints on Mercury's geodetic parameters via least squares minimization of crossover discrepancies. We simultaneously solve for orbital corrections for each MLA ground track, for the geodetic parameters of the International Astronomical Union‐recommended orientation model for Mercury (pole right‐ascension and declination coordinates, prime meridian rotation rate, and librations), and for the Mercury's Love number h2. We calibrate the formal errors of our solution based on closed‐loop simulations and on the level of robustness against a priori values, data selection, and parametrization. Our solution of the Mercury's rotational parameters is consistent with published values. In particular, our new estimate for the orientation of the pole places Mercury in a Cassini state, with an obliquity ϵ = 2.031 ± 0.03 arcmin compatible with previous “surface” related measurements. Moreover, we provide a first data‐based estimate of the Love number h2 = 1.55 ± 0.65. The latter is consistent with expectations from models of Mercury's interior, although its precision does not enable their refinement.

Augmenting NASA Europa Clipper by a small probe: Europa Tomography Probe (ETP) mission concept

The Jupiter's moon Europa is believed to be one of the few bodies in the Solar System able to host habitable environments. Compelling evidence for the existence of a subsurface ocean was provided by the magnetic field measurements carried out by the Galileo spacecraft.The NASA/JPL Europa Clipper mission is scheduled for launch in the early 2020s. Due to the harsh Jovian radiation environment and budget limitations, the initial concept of an orbiter was abandoned, and the spacecraft will perform only multiple flybys of the moon. Although Clipper will investigate the surface and subsurface properties with a powerful suite of instruments, such approach is not very favourable to a thorough investigation of Europa's deep interior structure.In 2015 the Europa Clipper project announced the possibility to carry an additional flight element with a mass of about 250 kg. We then started investigating a smallest mission aimed at regaining the original science objectives of Clipper for studying the deep interior structure of Europa. Although this additional mass is no longer available, we describe here a mission concept, based on a mothercraft-daughtercraft configuration, which could be of interest to future exploration missions. Our proposal is based on a small spacecraft, the Europa Tomography Probe (ETP), to be inserted in a polar orbit around Europa with a lifetime of a few months. ETP should carry only a magnetometer and a transponder for inter-satellite link, thus not foreseeing any direct link to ground. By exploiting the magnetic induction effect from Jupiter, the magnetometer will measure Europa's magnetic polarizability, thus providing information on the thickness and conductivity of the internal salty ocean, while the ETP transponder will enable a two-way relay link with the mother spacecraft during Europa flybys. The relay link will be used to transfer telemetry data and to carry out range rate measurements on board Clipper. This configuration minimizes the required power on ETP and provides high quality Doppler observables using an ultra-stable oscillator hosted on Clipper. The limiting noise in Doppler measurements will be the frequency stability of ETP and Clipper relay electronics. Doppler data allow the determination of Europa's static gravity field with high resolution along with its variable part due to eccentricity tides (Love number k2). Measurements of the moon's gravity also provide accurate data on its rotational state (e.g., Europa's obliquity and libration amplitude). Furthermore, the inter-satellite Doppler observables can improve the accuracy in the determination of Clipper's position relative to Europa at the level of a few meters in the radial direction. The precise positioning of the main spacecraft relative to Europa and the altimetric measurements from the radar REASON could be exploited to measure the tidal displacement (Love number h2) of the outer icy shell.The NASA's Europa Clipper is viewed as the first applicable study case for the mission concept presented in this paper, which is of general interest to a class of similar flyby missions that can be conceived not only for the Jovian system, but also for Saturn and the ice giants of the solar system, Uranus and Neptune. In the latter case, for example, a small orbiter of Triton could significantly boost the science return from a larger spacecraft focusing on the observation of the planet. In the case of Europa, we show that joint magnetic field, gravity, and possibly altimetric measurements, will place strong constraints on the deep interior structure of the moon and provide a good determination of the ice shell thickness and ocean depth.

Dual Frequency Orbiter-Radar System for the Observation of Seas and Tides on Titan: Extraterrestrial Oceanography from Satellite

Saturn’s largest moon, Titan, is believed to have a ~100 km thick ice shell above a global ocean of liquid water. Organic materials, including liquid hydrocarbon lakes and seas in its polar terrain, cover Titan’s surface, which makes it a world of two oceans. The RADAR instrument on board Cassini, was able to probe lakes and seas during few dedicated altimetric observations, revealing its capability to work as a sounder. Herein, we describe the design of, and scientific motivation for, a dual frequency X/Ka-band radar system that is able to investigate Titan’s subsurface liquid water ocean, as well as the depth and composition of its surface liquid hydrocarbon basins. The proposed system, which could take advantage of the telecommunications dish, can operate as a sounder, as Synthetic Aperture Radar (SAR) able to map the surface at tens meters of scale resolution, and when data are acquired from close-adjacent orbits, as a repeat-pass SAR interferometer (InSAR). The instrument, which is based on the architecture of the Cassini RADAR, can also characterize Titan’s interior by using geophysical measurements of the tidal amplitude to derive high accuracy estimates of the Love number h2 from a 1500 km circular orbit.

Viscoelastic Tides of Mercury and the Determination of its Inner Core Size

AbstractWe computed interior structure models of Mercury and analyzed their viscoelastic tidal response. The models are consistent with MErcury Surface, Space Environment, GEochemistry, and Ranging mission inferences of mean density, mean moment of inertia, moment of inertia of mantle and crust, and tidal Love number k2. Based on these constraints we predict the tidal Love number h2 to be in the range from 0.77 to 0.93. Using an Andrade rheology for the mantle the tidal phase‐lag is predicted to be 4° at maximum. The corresponding tidal dissipation in Mercury's silicate mantle induces a surface heat flux smaller than 0.16 mW/m2. We show that, independent of the adopted mantle rheological model, the ratio of the tidal Love numbers h2 and k2 provides a better constraint on the maximum inner core size with respect to other geodetic parameters (e.g., libration amplitude or a single Love number), provided it responds elastically to the solar tide. For inner cores larger than 700 km, and with the expected determination of h2 from the upcoming BepiColombo mission, it may be possible to constrain the size of the inner core. The measurement of the tidal phase‐lag with an accuracy better than ≈0.5° would further allow constraining the temperature at the core‐mantle boundary for a given grain size and therefore improve our understanding of the physical structure of Mercury's core.

The performance of the BepiColombo Laser Altimeter (BELA) prior launch and prospects for Mercury orbit operations

We explore the perspectives of the BepiColombo Laser Altimeter (BELA) onboard ESA / JAXA's Mercury mission BepiColombo and present an updated semi-analytical instrument performance model, in which we estimate signal-to-noise ratio, single shot probability of false detection, range errors and the accuracy of pulse width reconstruction. The model is generally applicable for other laser altimeters using matched filter algorithms for pulse detection and has been validated against the recently tested BELA flight model after integration on the BepiColombo spacecraft. Further, we perform numerical simulations of the instrument performance expected in orbit about Mercury. In particular, we study the measurement accuracy of topography, slopes and surface roughness, which will allow us to estimate local and global topographic coverage based on the current trajectory design. We also assess the potential for measuring the tidal Love number h2 using cross-over points, which we estimate to be constrained with an absolute accuracy of 0.14 corresponding to a relative accuracy of about 18% after two years in Mercury orbit.

Assessing the potential for measuring Europa's tidal Love number h2 using radar sounder and topographic imager data

The tidal Love number h2 is a key geophysical measurement for the characterization of Europa's interior, especially of its outer ice shell if a subsurface ocean is present. We performed numerical simulations to assess the potential for estimating h2 using altimetric measurements with a combination of radar sounding and stereo imaging data. The measurement principle exploits both delay and Doppler information in the radar surface return in combination with topography from a digital terrain model (DTM). The resulting radar range measurements at cross-over locations can be used in combination with radio science Doppler data for an improved trajectory solution and for estimating the h2 Love number. Our simulation results suggest that the absolute accuracy of h2 from the joint analysis of REASON (Radar for Europa Assessment and Sounding: Ocean to Near-surface) surface return and EIS (Europa Imaging System) DTM data will be in the range of 0.04–0.17 assuming full radio link coverage. The error is controlled by the SNR budget and DTM quality, both dependent on the surface properties of Europa. We estimate that this would unambiguously confirm (or reject) the global ocean hypothesis and, in combination with a nominal radio-science based measurement of the tidal Love number k2, constrain the thickness of Europa's outer ice shell to up to ±15 km.

Measuring tidal deformations by laser altimetry. A performance model for the Ganymede Laser Altimeter

Invaluable information about the interior of icy satellites orbiting close to the giant planets can be gained by monitoring the response of the satellite's surfaces to external tidal forces. Due to its geodetic accuracy, laser altimetry is the method of choice to measure time-dependent radial surface displacements from orbit. We present an instrument performance model with special focus on the capabilities to determine the corresponding tidal Love number h2 and apply the model to the Ganymede Laser Altimeter (GALA) on board of the Jupiter Icy Moons Explorer (JUICE). Based on the instrument and spacecraft performance, we derive the range error and the measurement capabilities of the GALA instrument to determine the amplitude of the tide induced radial displacement of Ganymede's surface using the cross-over technique. We find that h2 of Ganymede can be determined with an accuracy of better than 2% by using data acquired during the nominal mission. Furthermore, we show that this accuracy is sufficient to confirm the presence of a putative subsurface water ocean and, additionally, to constrain the thickness of the overlaying ice shell to ±20km.

Simulated recovery of Europa's global shape and tidal Love numbers from altimetry and radio tracking during a dedicated flyby tour

AbstractThe fundamental scientific objectives for future spacecraft exploration of Jupiter's moon Europa include confirmation of the existence of subsurface ocean beneath the surface ice shell and constraints on the physical properties of the ocean. Here we conduct a comprehensive simulation of a multiple‐flyby mission. We demonstrate that radio tracking data can provide an estimate of the gravitational tidal Love number k2 with sufficient precision to confirm the presence of a liquid layer. We further show that a capable long‐range laser altimeter can improve determination of the spacecraft position, improve the k2 determination (<1% error), and enable the estimation of the planetary shape and Love number h2 (3–4% error), which is directly related to the amplitude of the surface tidal deformation. These measurements, in addition to the global shape accurately constrained by the long altimetric profiles, can yield further constraints on the interior structure of Europa.

Detection of the lunar body tide by the Lunar Orbiter Laser Altimeter.

The Lunar Orbiter Laser Altimeter instrument onboard the Lunar Reconnaissance Orbiter spacecraft collected more than 5 billion measurements in the nominal 50 km orbit over ∼10,000 orbits. The data precision, geodetic accuracy, and spatial distribution enable two-dimensional crossovers to be used to infer relative radial position corrections between tracks to better than ∼1 m. We use nearly 500,000 altimetric crossovers to separate remaining high-frequency spacecraft trajectory errors from the periodic radial surface tidal deformation. The unusual sampling of the lunar body tide from polar lunar orbit limits the size of the typical differential signal expected at ground track intersections to ∼10 cm. Nevertheless, we reliably detect the topographic tidal signal and estimate the associated Love number h2 to be 0.0371 ± 0.0033, which is consistent with but lower than recent results from lunar laser ranging.Key PointsAltimetric data are used to create radial constraints on the tidal deformationThe body tide amplitude is estimated from the crossover dataThe estimated Love number is consistent with previous estimates but more precise

Estimation of the Love and Shida numbers: h2, l2 using SLR data for the low satellites

In this paper we present results for the global elastic parameters: Love number h2 and Shida number l2 derived from the analysis of Satellite Laser Ranging (SLR) data. SLR data for the two low satellites STELLA (H=800km) and STARLETTE (H=810km) observed during 2.5years from January 3, 2005 until July 1, 2007 with 18 globally distributed ground stations were analyzed. The analysis was done separately for the two satellites. We do a sequential analysis and study the stability and convergence of the estimates as a function of length of the data set used.The final adjusted values for h2 equal to 0.6163±0.0037 and 0.6048±0.0025, and those for l2 equal to 0.0176±0.0017 and 0.1151±0.0010 for STELLA and STARLETTE tracking data are compared to estimates we previously published based on data for two high satellites LAGEOS1 (H=5860km) and LAGEOS2 (H=5620km) (Rutkowska and Jagoda, 2010a). A major discrepancy between the two solutions was found for the Shida number l2.

Dissipation at tidal and seismic frequencies in a melt‐free Moon

We calculate viscoelastic dissipation in the Moon using a rheological (extended Burgers) model based on laboratory deformation of melt‐free polycrystalline olivine. Lunar temperature structures are calculated assuming steady state conduction with variable internal heat production and core heat flux. Successful models can reproduce the dissipation factor (Q) measured at both tidal and seismic frequencies, and the tidal Love numbers h2 and k2, without requiring any mantle melting. However, the frequency‐dependence of our modelQat tidal periods has the opposite sign to that observed. Using the apparently unrelaxed nature of the core‐mantle boundary (CMB), the best fit models require mantle grain sizes of ∼1 cm and CMB temperatures of ≈1700 K. If melt or volatiles are present, the lunar temperature structure must be colder than our melt‐free models. We estimate a present‐day mantle heat production rate of 9–10 nWm−3, suggesting that roughly half of the Moon's radiogenic elements are in the crust.

High precision gravity measurements with the dual sphere superconducting gravimeter in Sutherland (South Africa)

Since March 2000, the reconstructed Dual Sphere Superconducting Gravimeter (SG) with two gravity sensors is continuously recording gravity data at the newly constructed South African Geodynamic Observatory Sutherland of GFZ (SAGOS). Additionally, meteorological sensors and a ground water table sensor were installed for estimation of gravity changes induced by the atmosphere and hydrosphere. The observatory is designed for high precision geodynamic observations of the Earth with space techniques and ground instruments. The Noise Magnitude characterising the quality of the site Sutherland is close and even smaller than the “New Low Noise Model” values at frequencies below 1 mHz. This result characterises the excellent quality of the site for the observation of weak gravity signals. For calibration of the SG gravity sensors, absolute gravity measurements have been carried out at SAGOS. The calibration factor could be fixed with an accuracy of 0.2%. Tidal amplitudes, amplitude factors and phase leads have been determined for Sutherland with high resolution and accuracy for the first time. After applying ocean loading correction for the main diurnal and semidiurnal waves, there are still discrepancies between the measured and model tidal parameters. These discrepancies have to be minimised by improving the models and data correction for non-tidal induced gravity effects. The maximal vertical surface shift caused by the earth tides and mass redistribution in the atmosphere during the time from March 27th 2000 to August 1st 2001 has been determined to 41.9 cm by using the Love numbers h2 and k2 derived from the tidal and deformation potential. The data of the earthquake near the coast of Peru on June 23rd 2001 have been analyzed for detection of the free oscillation modes of the Earth. Especially the long periodic modes 0S3, 2S1 and 0S2 are very well marked after this earthquake and less disturbed because of the low noise site. They confirm the model frequencies for these modes.

The synthetical elastic characteristic parameters of earth tide and its abnormal precursor evolution

This paper intends to analyze synthetically a physical property of the crust using tidal observations. We regarded Love number that can describe both micro deformation and elasticity of the crust in tidal observation as a physical and mechanical parameter. Synthetically inverse and calculate Love number on the earth surface using actual tide data included tidal linear strain, plane strain, volumetric strain, tilt, gravity and water level. Research seismic mechanical process and its precursory abnormality of a region based on the variation of Love number. In this paper, a feasible way and method to synthetically calculate the second-step Love number on the earth surface is discussed and determined by tidal theory. Love number h 2 in the boundary region of Sichuan and Yunnan province is practically calculated using tidal observations, and the precursory anomalous variation of the Lijiang earthquake (M s=7.0, February 9, 1996) is analyzed. The result shows that large-scale Love number anomaly in the Sichuan-Yunnan region began from 1994. The anomaly moves from south to southeast, north and northwest in the region, and the epicenter is an anomalous empty. Finally, the anomalous area was concentrated to Lijiang and Yongsheng located northwestern Sichuan-Yunnan region on December 1995, and the epicenter area in future is drawn out clearly.

Geometrical determination of the Love Number h2 at four tidal frequencies

The Love number h2 is determined at the M2, N2, O1, and K1 frequencies by combining satellite altimetry with in situ ocean tidal measurements. Topex/Poseidon altimetry is used to measure the (ocean + earth) tide, while gauges at 114 island and ocean‐bottom sites determine the ocean tide; the difference sets constraints on h2. Our h2 standard errors are somewhat larger than recent VLBI estimates, but they clearly delineate the resonance in K1 at the 6‐σ level.

Measurement of diurnal and semidiurnal rotational variations and tidal parameters of Earth

We discuss the determination of diurnal and semidiurnal variations in the rotation rate and the direction of rotation axis of Earth from the analysis of 8 years of very long baseline interferometry (VLBI) data. This analysis clearly show that these variations are largely periodic and tidally driven; that is, the periods of the variations correspond to the periods of the largest lunar and solar tides. For rotation rate variations, expressed in terms of changes in universal time (UT), the tidal lines with the largest observed signals are O1 (amplitude 23.5 microseconds in time (μs), period 25.82 solar hours); K1 (18.9 μs, 23.93 hours); M2 (17.9 μs, 12.54 hours); and S2 (8.6 μs, 12.00 hours). For variations in the direction of the rotation axis (polar motion), significant signals exist in the retrograde semidiurnal band at the M2 and S2 tides (amplitudes 265 and 119 microarc seconds (μas), respectively); the prograde diurnal band at the O1, K1, and P1 tides (amplitudes 199,152, and 60 μas, respectively); and the prograde semidiurnal band at the M2 and K2 tides (amplitudes 58 and 39 μas, respectively). Variations in the retrograde diurnal band are represented by corrections to the nutations of Earth's body axes, and our estimates here are consistent with previous estimates except that a previously noted discrepancy in the 13.66‐day nutation (corresponding to the O1 tide) is largely removed in this new analysis. We estimate that the standard deviations of these estimates are 1.0 μs for the UTl variations and 14–16 μas for the polar motion terms. These uncertainties correspond to surface displacements of ∼0.5 mm. From the analysis of atmospheric angular momentum data we conclude that variations in UTl excited by the atmosphere with subdaily periods are small (∼1 μs). We find that the average radial tidal displacements of the VLBI sites in the diurnal band are largely consistent with known deficiencies in current tidal models, i.e., deficiencies of up to 0.9 mm in the treatment of the free core nutation resonance. In the semidiurnal band, our analysis yields estimates of the second‐degree harmonic radial Love number h2 at the M2 tide of 0.604+i0.005±0.002. The most likely explanation for the rotational variations are the effects of ocean tides, but there may also be some contributions from atmospheric tides, the effects of triaxiality of Earth, and the equatorial second‐degree‐harmonic components of the core‐mantle boundary.

Temporal variations of rotational origin in the absolute value of gravity

When correcting precise gravity measurements for polar motion, the Earth's rotational deformation must be considered, as this will increase the correction based on a rigid Earth by about 15%. Conversely the gravity observations can be used to estimate the Love numbers h2 and k2 at the Chandler frequency.