Love Number Research Articles

Theoretical viscoelastic deformations due to an expansion source in a spherical earth model

SUMMARYSources of expansion, such as nuclear explosions and magma chambers, can cause various surface effects, such as displacement and gravity field change on the Earth. The deformation simulation of an expansion source can be traced back to certain classic works nearly 70 yr ago, which were primarily based on a half-space earth model. However, the deformation of expansion sources in a spherical earth model has not been presented in the context of the semi-analytical viscoelastic Love numbers framework. In this study, we present the deformation Green's function for an expansion source in a spherical, non-rotating, viscoelastic, isotropic layered earth model using a semi-analytical method. We further obtain the analytical Green's functions for an elastic model in the near field. In addition, the influence of the stratification, curvature, and viscosity on the deformation is analysed using a layered spherical earth model. The suitability and effectiveness of the formulas are verified by numerical experiments and compared with previous results. The method presented in this paper is useful for explaining the viscoelastic deformations caused by the deep expansion source, for example, large magma chambers and upwelling of a mantle plume, in the layered viscoelastic earth model.

A Recipe for the Geophysical Exploration of Enceladus

Abstract Orbital geophysical investigations of Enceladus are critical to understanding its energy budget. In this paper, we identified key science questions for the geophysical exploration of Enceladus, answering which would support future assessment of Enceladus’ astrobiological potential. Using a Bayesian framework, we explored how science requirements map to measurement requirements. We performed mission simulations to study the sensitivity of single-spacecraft and dual-spacecraft configurations to static gravity and tidal Love numbers of Enceladus. We find that mapping Enceladus’ gravity field, improving the accuracy of the physical libration amplitude, and measuring Enceladus’ tidal response would provide critical constraints on the internal structure and establish a framework for assessing Enceladus’ long-term habitability. This kind of investigation could be carried out as part of a life search mission at little additional resource requirements.

Tidal deformability of dressed black holes and tests of ultralight bosons in extended mass ranges

The deformability of a compact object under the presence of a tidal perturbation is encoded in the tidal Love numbers (TLNs), which vanish for isolated black holes in vacuum.We show that the TLNs of black holes surrounded by matter fields do not vanish and can be used to probe the environment around binary black holes. In particular, we compute the TLNs for the case of a black hole surrounded by a scalar condensate under the presence of scalar and vector tidal perturbations, finding a strong power-law behavior of the TLN in terms of the mass of the scalar field. Using this result as a proxy for gravitational tidal perturbations, we show that future gravitational-wave detectors like the Einstein Telescope and LISA can impose stringent constraints on the mass of ultralight bosons that condensate around black holes due to accretion or superradiance. Interestingly, LISA could measure the tidal deformability of dressed black holes across the range from stellar-mass (≈ 102 M ☉) to supermassive (≈ 107 M ☉) objects, providing a measurement of the mass of ultralight bosons in the range (10-17 – 10-13) eV with less than 10% accuracy, thus filling the gap between other superradiance-driven constraints coming from terrestrial and space interferometers. Altogether, LISA and Einstein Telescope can probe tidal effects from dressed black holes in the combined mass range (10-17 – 10-11) eV.

Tidal deformation and dissipation of rotating black holes

We show that rotating black holes do not experience any tidal deformation when they are perturbed by a weak and adiabatic gravitational field. The tidal deformability of an object is quantified by the so-called "Love numbers", which describe the object's linear response to its external tidal field. In this work, we compute the Love numbers of Kerr black holes and find that they vanish identically. We also compute the dissipative part of the black hole's tidal response, which is non-vanishing due to the absorptive nature of the event horizon. Our results hold for arbitrary values of black hole spin, for both the electric-type and magnetic-type perturbations, and to all orders in the multipole expansion of the tidal field. The boundary conditions at the event horizon and at asymptotic infinity are incorporated in our study, as they are crucial for understanding the way in which these tidal effects are mapped onto gravitational-wave observables. In closing, we address the ambiguity issue of Love numbers in General Relativity, which we argue is resolved when those boundary conditions are taken into account. Our findings provide essential inputs for current efforts to probe the nature of compact objects through the gravitational waves emitted by binary systems.

Jupiter’s Dynamical Love Number

Abstract Recent observations by the Juno spacecraft have revealed that the tidal Love number k 2 of Jupiter is 4% lower than the hydrostatic value. We present a simple calculation of the dynamical Love number of Jupiter that explains the observed “anomaly.” The Love number is usually dominated by the response of the (rotation-modified) f-modes of the planet. Our method also allows for efficient computation of high-order dynamical Love numbers. While the inertial-mode contributions to the Love numbers are negligible, a sufficiently strong stratification in a large region of the planet’s interior would induce significant g-mode responses and influence the measured Love numbers.

Possible Deep Structure and Composition of Venus With Respect to the Current Knowledge From Geodetic Data

AbstractAlmost no in situ seismic data are available for Venus; therefore, we can constrain the deep structure of this planet only from the geodetic data obtained by spacecraft. Of particular interest is Venus’s core‐mantle boundary, which holds clues about the origin and evolution of this celestial body. In this study, we build a series of different mantle/core models of Venus based on several mantle composition models and compare their associated Love numbers k2 with the observed values. Due to the large uncertainty in the observed values of k2, the state of Venus’s core cannot be reliably constrained. However, the expected precision of k2 obtained by the EnVision mission will sufficiently reduce the acceptable model space and contribute to estimating the mantle viscosity structure. Based on current geodetic data, we find that the bottom of Venus’s mantle may not feature a phase transition from perovskite to post‐perovskite if the FeO content of the mantle is less than 8.1 wt%; this region may be different from Earth’s D” layer. Furthermore, we find that the combination of observed k2 and Q can be used to distinguish whether the lowermost part of Venus’s mantle is a high‐temperature basalt layer or a thin thermal boundary layer if observations of Q can be obtained.

Isostasy with Love: II Airy compensation arising from viscoelastic relaxation

SUMMARY In modern geodynamics, isostasy can be viewed either as the static equilibrium of the crust that minimizes deviatoric stresses, or as a dynamic process resulting from the viscous relaxation of the non-hydrostatic crustal shape. Paper I gave a general formulation of Airy isostasy as an elastic loading problem solved with Love numbers, and applied it to the case of minimum stress isostasy. In this sequel, the same framework is used to study Airy isostasy as the long-time evolution of a viscoelastic shell submitted to surface and internal loads. Isostatic ratios are defined in terms of time-dependent deviatoric Love numbers. Dynamic isostasy depends on the loading history, two examples of which are the constant load applied on the surface in the far past and the constant shape maintained by addition or removal of material at the compensation depth. The former model results in a shape decreasing exponentially with time and has no elastic analogue, whereas the latter (stationary) model is equivalent to a form of elastic isostasy. Viscoelastic and viscous approaches are completely equivalent. If both load and shape vary slowly with time, isostatic ratios look like those of the stationary model. Isostatic models thus belong to two independent groups: the elastic/stationary approaches and the time-dependent approaches. If the shell is homogeneous, all models predict a similar compensation of large-scale gravity perturbations. If the shell rheology depends on depth, stationary models predict more compensation at long wavelengths, whereas time-dependent models result in negligible compensation. Mathematica and Fortran codes are available for computing the isostatic ratios of an incompressible body with three homogeneous layers.

Astrophysical and Theoretical Physics Implications from Multimessenger Neutron Star Observations.

The Neutron Star Interior Composition Explorer (NICER) recently measured the mass and equatorial radius of the isolated neutron star PSR J0030+0451. We use these measurements to infer the moment of inertia, the quadrupole moment, and the surface eccentricity of an isolated neutron star for the first time, using relations between these quantities that are insensitive to the unknown equation of state of supranuclear matter. We also use these results to forecast the moment of inertia of neutron star A in the double pulsar binary J0737-3039, a quantity anticipated to be directly measured in the coming decade with radio observations. Combining this information with the measurement of the tidal Love number with LIGO/Virgo observations, we propose and implement the first theory-agnostic and equation-of-state-insensitive test of general relativity. Specializing these constraints to a particular modified theory, we find that consistency with general relativity places the most stringent constraint on gravitational parity violation to date, surpassing all other previously reported bounds by 7 orders of magnitude and opens the path for a future test of general relativity with multimessenger neutron star observations.

Influence of Low Europa Orbit design on gravity field recovery

The characterization of Europa's surface and interior ocean and ice shell is key to explore the habitability of the fourth largest moon of Jupiter. Europa Clipper will largely contribute to our knowledge of Europa with multiple dedicated flybys. However, global mapping can only be significantly improved by means of an orbiter with high inclination and low altitude, as it will be done in the case of Ganymede by JUICE. This was the strategy of several mission proposals (Joint Europa Mission, Jupiter Europa Orbiter, HADES). A specific difficulty in designing suitable orbits in the Jovian system is related to the orbit stability, which is highly impacted by the influence of Jupiter as a third body exerting strong perturbations on a Europa orbiter. This results in constraints for potential science orbits which would allow, e.g., for the determination of Europa's gravity field using radio tracking to the probe. In this presentation, we will analyze a set of different low Europa orbits and compare their scientific value for gravity field recovery. We will explore different orbital characteristics, with the possibility to use repetitive ground track orbits. The analysis will be performed based on closed-loop simulations using the planetary extension of the Bernese GNSS Software. We simulate realistic Doppler tracking data (2-way X-band Doppler range rate) from the Deep Space Network. These observations are then used to reconstruct the orbit as well as geodetic parameters, such as Europa gravity field parameters and the tidal Love number k2. We will discuss the quality of the retrieved solutions.

On the vanishing of Love numbers for Kerr black holes

It was shown recently that the static tidal response coefficients, called Love numbers, vanish identically for Kerr black holes in four dimensions. In this work, we confirm this result and extend it to the case of spin-0 and spin-1 perturbations. We compute the static response of Kerr black holes to scalar, electromagnetic, and gravitational fields at all orders in black hole spin. We use the unambiguous and gauge-invariant definition of Love numbers and their spin-0 and spin-1 analogs as Wilson coefficients of the point particle effective field theory. This definition also allows one to clearly distinguish between conservative and dissipative response contributions. We demonstrate that the behavior of Kerr black hole responses to spin-0 and spin-1 fields is very similar to that of the spin-2 perturbations. In particular, static conservative responses vanish identically for spinning black holes. This implies that vanishing Love numbers are a generic property of black holes in four-dimensional general relativity. We also show that the dissipative part of the response does not vanish even for static perturbations due to frame-dragging.

Determination of Venus’ Interior Structure with EnVision

The Venusian geological features are poorly gravity-resolved, and the state of the core is not well constrained, preventing an understanding of Venus’ cooling history. The EnVision candidate mission to the ESA’s Cosmic Vision Programme consists of a low-altitude orbiter to investigate geological and atmospheric processes. The gravity experiment aboard this mission aims to determine Venus’ geophysical parameters to fully characterize its internal structure. By analyzing the radio-tracking data that will be acquired through daily operations over six Venusian days (four Earth’s years), we will derive a highly accurate gravity field (spatial resolution better than ~170 km), allowing detection of lateral variations of the lithosphere and crust properties beneath most of the geological features. The expected 0.3% error on the Love number k2, 0.1° error on the tidal phase lag and 1.4% error on the moment of inertia are fundamental to constrain the core size and state as well as the mantle viscosity.

Post-seismic crustal internal deformation in a layered earth model

SUMMARY This study introduces a novel method for computing post-seismic crustal internal deformation in a layered earth model. The surface dislocation Love number (DLN) calculated by the reciprocity theorem was implemented as the initial value. Furthermore, numerical integration of the value from the Earth's surface to the interior was undertaken to obtain the internal DLN. This method does not require a combination of the general solution and particular solution for the calculation of internal deformation above the seismic source, thus avoiding the loss of precision. When the post-seismic deformation within a certain period is calculated, the particular solutions at the beginning and end of the considered period cancel each other. This simplifies the calculation of post-seismic internal deformation. The numerical results depict that as the degrees increase, the post-seismic DLN reaches stability in a shorter interval of time. Thus, for improved efficiency of the post-seismic internal deformation calculation, the post-seismic DLNs should be calculated within 2000 degree and integrated with the coseismic results. As an application, the post-seismic Coulomb failure stress changes (∆CFS) induced by the 2011 Tohoku-Oki earthquake in the near field around the Japanese archipelagos and two major faults in Northeast China were simulated. The results exhibit that the ∆CFS values in the near field agree well with those simulated by the method in a half-space layered earth model, thus verifying the present method. The coseismic ∆CFS on the Mishan-Dunhua fault in Northeast China, as an example, is only 0.094–0.668 KPa. However, the ∆CFS caused by the viscoelastic relaxation of the mantle within 5 yr following the 2011 Tohoku-Oki event on the same fault exceeds the coseismic results. Therefore, the cumulative effect of the viscoelastic relaxation of the mantle is deserving of attention.

Tidal Love numbers of Kerr black holes

The open question of whether a Kerr black hole can become tidally deformed or not has profound implications for fundamental physics and gravitational-wave astronomy. We consider a Kerr black hole embedded in a weak and slowly varying, but otherwise arbitrary, multipolar tidal environment. By solving the static Teukolsky equation for the gauge-invariant Weyl scalar ${\ensuremath{\psi}}_{0}$ and by reconstructing the corresponding metric perturbation in an ingoing radiation gauge, for a general harmonic index $\ensuremath{\ell}$, we compute the linear response of a Kerr black hole to the tidal field. This linear response vanishes identically for a Schwarzschild black hole and for an axisymmetric perturbation of a spinning black hole. For a nonaxisymmetric perturbation of a spinning black hole, however, the linear response does not vanish, and it contributes to the Geroch-Hansen multipole moments of the perturbed Kerr geometry. As an application, we compute explicitly the rotational black hole tidal Love numbers that couple the induced quadrupole moments to the quadrupolar tidal fields, to linear order in the black hole spin, and we introduce the corresponding notion of a tidal Love tensor. Finally, we show that those induced quadrupole moments are closely related to the well-known physical phenomenon of tidal torquing of a spinning body interacting with a tidal gravitational environment.

Static response and Love numbers of Schwarzschild black holes

We derive the quadratic action for the physical degrees of freedom of massless spin-0, spin-1, and spin-2 perturbations on a Schwarzschild-(A)dS background in arbitrary dimensions. We then use these results to compute the static response of asymptotically flat Schwarzschild black holes to external fields. Our analysis reproduces known facts about black hole Love numbers — in particular that they vanish for all types of perturbation in four spacetime dimensions — but also leads to new results. For instance, we find that neutral Schwarzschild black holes polarize in the presence of an electromagnetic background in any number of spacetime dimensions except four. Moreover, we calculate for the first time black hole Love numbers for vector-type gravitational perturbations in higher dimensions and find that they generically do not vanish. Along the way, we shed some light on an apparent discrepancy between previous results in the literature, and clarify some aspects of the matching between perturbative calculations of static response on a Schwarzschild background and the point-particle effective theory.

The Thermo‐Chemical Evolution of Mars With a Strongly Stratified Mantle

AbstractThe Martian mantle probably experienced an early global magma ocean stage. The crystallization and the fractionation and overturn of such a magma ocean likely led to the formation of a compositionally distinct layer at the bottom of the mantle. This layer would have been heavily enriched in iron and in heat‐producing elements (HPE). The significant iron enrichment can lead to long‐term stability with little mixing between the layer and the overlying mantle. We studied the influence of such an enriched basal layer on the thermal and chemical evolution of the Martian mantle using both 2‐D finite‐volume modeling at mantle scale, and a parameterized convection approach at the entire planetary scale. The basal layer is most likely stably stratified because of its moderate thickness and/or its gradual enrichment in iron with depth that prevents the development of convection in this region. We explored a wide parameter space in our parameterized models, including the layer thickness and the mantle rheology. We show that the presence of an enriched basal layer has a dramatic influence on the thermo‐chemical evolution of Mars, strongly delaying deep cooling, and significantly affecting nearly all present‐day characteristics of the planet (heat flux, thermal state, crustal and lithospheric thickness, Love number and tidal dissipation). In particular, the enrichment of the layer in iron and HPE generates large volumes of stable melt near the core‐mantle boundary. Due to their intrinsic low viscosity and seismic velocities, these regions of silicate melt could be erroneously interpreted as core material.

Dynamical Tides in Jupiter as Revealed by Juno

Abstract The Juno orbiter has continued to collect data on Jupiter's gravity field with unprecedented precision since 2016, recently reporting a nonhydrostatic component in the tidal response of the planet. At the mid-mission perijove 17, Juno registered a Love number k 2 = 0.565 ± 0.006 that is −4% ± 1% (1σ) from the theoretical hydrostatic k 2 ( hs ) = 0.590 . Here we assess whether the aforementioned departure of tides from hydrostatic equilibrium represents the neglected gravitational contribution of dynamical tides. We employ perturbation theory and simple tidal models to calculate a fractional dynamical correction Δk 2 to the well-known hydrostatic k 2. Exploiting the analytical simplicity of a toy uniform-density model, we show how the Coriolis acceleration motivates the negative sign in the Δk 2 observed by Juno. By simplifying Jupiter’s interior into a coreless, fully convective, and chemically homogeneous body, we calculate Δk 2 in a model following an n = 1 polytrope equation of state. Our numerical results for the n = 1 polytrope qualitatively follow the behavior of the uniform-density model, mostly because the main component of the tidal flow is similar in each case. Our results indicate that the gravitational effect of the Io-induced dynamical tide leads to Δk 2 = − 4% ± 1%, in agreement with the nonhydrostatic component reported by Juno. Consequently, our results suggest that Juno obtained the first unambiguous detection of the gravitational effect of dynamical tides in a gas giant planet. These results facilitate a future interpretation of Juno tidal gravity data with the purpose of elucidating the existence of a dilute core in Jupiter.

Time-space characteristics of viscoelastic post-seismic deformations corresponding to different rheology models

On a long time (> 1 a) scale, the viscoelastic properties of mantle media significantly affect post-seismic deformation. The stress field disturbance in viscoelastic medium caused by fault slip gradually relax, and the relaxation process and its temporal-spatial characteristics are determined by the viscoelastic model. In this paper, we assume that the mantle media are types of common linear rheological models, i.e., the Burgers body, the standard linear solid, and the Maxell body, and we calculate the dislocation Love number and Green function for a spherically symmetric, non-rotating, viscoelastic, and isotropic (SNRVEI) Earth model. The characteristics of the post-seismic relaxation deformations corresponding to the different models are compared. Our results show that for a short time period, the Burgers body and standard linear solid are similar; while for the long time period, the Burgers body and Maxwell body are similar. This suggests that the observations of post-seismic deformation on the surface have a great potential for the inversion of underground viscoelastic structures. However, the potential of using surface displacement to distinguish different rheological models is limited when the observation period is not long enough.

Universal relationship for magnetized white dwarfs

Recently super-Chandrasekhar mass limit has been derived theoretically in presence of strong magnetic field to complement experimental observations. In the framework of Newtonian physics, we have studied the equilibrium configurations of such magnetized white dwarfs by using the relativistic Thomas-Fermi equation of state for magnetized white-dwarfs. Hartle formalism, for slowly rotating stars, has been employed to obtain the equations of equilibrium. Various physical quantities of uniformly rotating and non-rotating white dwarfs have been calculated within this formalism. Consequently, the universality relationship between the moment of inertia(I), rotational love number($\lambda$) and spin induced quadrupole moment(Q), namely the I-Love-Q relationship, has been investigated for such magnetized white dwarfs. The relationship between I, eccentricity and Q i.e. I-eccentricity-Q relationship has also been derived. Further, we have found that, the I-eccentricity-Q relationship is more universal in comparison to I-Love-Q relationship.

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.

Secular changes in length of day: Effect of the mass redistribution

In this paper the secular change in the length of day due to mass redistribution effects is revisited using the Hamiltonian formalism of the Earth rotation theories. The framework is a two-layer deformable Earth model including dissipative effects at the core–mantle boundary, which are described through a coupling torque formulated by means of generalized forces. The theoretical development leads to the introduction of an effective time-averaged polar inertia moment, which allows us to quantify the level of core–mantle coupling throughout the secular evolution of the Earth. Taking advantage of the canonical procedure, we obtain a closed analytical formula for the secular deceleration of the rotation rate, numerical evaluation of which is performed using frequency-dependent Love numbers corresponding to solid and oceanic tides. With this Earth modeling, under the widespread assumption of totally coupled core and mantle layers in the long term response, a secular angular acceleration of − 1328.6′′ cy−2 is obtained, which is equivalent to an increase of 2.418 ms cy−1 in the length of day. The ocean tides and the semidiurnal band of the mass-redistribution-perturbing potential, mostly induced by the Moon, constitute the main part of this deceleration. This estimate is shown to be in very good agreement with recent observational values, and with other theoretical predictions including comparable modeling features.