Love Number Research Articles

I−Love−C relation for an anisotropic neutron star

One of the most common assumptions has been made that the pressure inside the star is isotropic in nature. However, the pressure is locally anisotropic in nature which is a more realistic case. In this study, we investigate certain properties of anisotropic neutron stars with the scalar pressure anisotropy model. Different perfect fluid conditions are tested within the star with the relativistic mean-field model equation of states (EOSs). The anisotropic neutron star properties such as mass ($M$), radius ($R$), compactness ($C$), Love number ($k_2$), dimensionless tidal deformability ($\Lambda$), and the moment of inertia ($I$) are calculated. The magnitude of the quantities as mentioned above increases (decreases) with the positive (negative) value of anisotropy except $k_2$ and $\Lambda$. The Universal relation $I-$Love$-C$ is calculated with almost 58 EOSs spans from relativistic to non-relativistic cases. We observed that the relations between them get weaker when we include anisotropicity. With the help of the GW170817 tidal deformability limit and radii constraints from different approaches, we find that the anisotropic parameter is less than 1.0 if one uses the BL model. Using the universal relation and the tidal deformability bound given by the GW170817, we put a theoretical limit for the canonical radius, $R_{1.4}=10.74_{-1.36}^{+1.84}$ km, and the moment of inertia, $I_{1.4} = 1.77_{-0.09}^{+0.17}\times10^{45}$ g cm$^2$ with 90\% confidence limit for isotropic stars. Similarly, for anisotropic stars with $\lambda_{\rm BL}=1.0$, the values are $R_{1.4}=11.74_{-1.54}^{+2.11}$ km, $I_{1.4} = 2.40_{-0.08}^{+0.17} \times10^{45}$ g cm$^2$ respectively.

Tidal Deformability of Neutron Stars in Unimodular Gravity

Unimodular gravity is a modified theory with respect to general relativity by an extra condition that the determinant of the metric is fixed. Especially, if the energy-momentum tensor is not imposed to be conserved separately, a new geometric structure will appear with potential observational signatures. In this paper, we study the tidal deformability of a compact star in unimodular gravity under the assumption of a non-conserved energy-momentum tensor. Both the electric-type and magnetic-type quadrupole tidal Love numbers are calculated for neutron stars using the polytrope model. It is found that the electric-type tidal Love numbers are monotonically increasing, but the magnetic-type ones are decreasing, with the increase in the non-conservation parameter. Compared with the observational data from the detected gravitational-wave events, a small negative non-conservation parameter is favored.

Love symmetry

Perturbations of massless fields in the Kerr-Newman black hole background enjoy a (“Love”) SL(2, ℝ) symmetry in the suitably defined near zone approximation. We present a detailed study of this symmetry and show how the intricate behavior of black hole responses in four and higher dimensions can be understood from the SL(2, ℝ) representation theory. In particular, static perturbations of four-dimensional black holes belong to highest weight SL(2, ℝ) representations. It is this highest weight properety that forces the static Love numbers to vanish. We find that the Love symmetry is tightly connected to the enhanced isometries of extremal black holes. This relation is simplest for extremal charged spherically symmetric (Reissner-Nordström) solutions, where the Love symmetry exactly reduces to the isometry of the near horizon AdS2 throat. For rotating (Kerr-Newman) black holes one is lead to consider an infinite-dimensional SL(2, ℝ) ⋉ hat{textrm{U}}{(1)}_{mathcal{V}} extension of the Love symmetry. It contains three physically distinct subalgebras: the Love algebra, the Starobinsky near zone algebra, and the near horizon algebra that becomes the Bardeen-Horowitz isometry in the extremal limit. We also discuss other aspects of the Love symmetry, such as the geometric meaning of its generators for spin weighted fields, connection to the no-hair theorems, non-renormalization of Love numbers, its relation to (non-extremal) Kerr/CFT correspondence and prospects of its existence in modified theories of gravity.

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.

Transient Green's functions of dislocations in transversely isotropic and layered poroelastic half-spaces

We derive, for the first time, the transient response (or Green's function, GF) induced by a general point dislocation in a transversely isotropic and layered poroelastic half-space where the contributions from the fluid dislocation and fluid-phase coupling effect are considered. The GF is expressed in terms of the powerful Fourier-Bessel series system of vector functions recently introduced (with the expansion coefficients being the novel dislocation Love numbers). The corresponding source functions, i.e. discontinuities across the source level, are derived, which show that 1) the fluid-phase creates a new type of source functions called fluid dislocation, and 2) it further contributes directly to the traditional solid horizontal tensile dislocation (or tensile-fracture) via the Biot effective stress coefficient. While the dual-variable and position (DVP) method is applied to take care of multilayers, the Talbot's method is employed to carry out the inverse Laplace transform, both showing excellent numerical stability, efficiency, and accuracy. Key features of these GFs are analyzed numerically. It is shown that 1) the poroelastic process is featured by some transient statuses, including drained and undrained limits; 2) while these two limits are sharply different when the dislocation source is a vertical strike-slip or horizontal tensile-fracture, they are the same when a vertical dip-slip or vertical tensile-fracture is in a homogeneous half-space; 3) in all the cases, there are temporal poroelastic deformations which are significantly different from these two limits; and 4) the fluid dislocation alone could significantly contribute to the poroelastic deformation at the earlier time history. These GFs provide the kernel functions in the corresponding boundary element formulation and the method of fundamental solutions, with potential applications in geomechanics and biomedical engineering.

Effects of Earth's gravitation and compressibility on co- and post-seismic deformations

SUMMARY In this paper, we systematically analyse the effect of Earth's gravitation and compressibility on co- and post-seismic deformations. Based on the analytical solution of dislocation Love number (DLN) in a non-gravitational and incompressible homogeneous Earth model, it was seen that reducing the density of the homogeneous Earth model can provide results equivalent to a non-gravitational model while expanding the Lamé constant of the homogeneous model gives results equivalent to those from an incompressible model. Then, we applied a non-gravitational and incompressible treatment in a layered spherical Earth model, and calculated the co- and post-seismic Green's functions for four independent sources. It is found that the effect of Earth's gravitation and compressibility are dependent on source depth and time after the event. The effect of gravitation on the co-seismic Green's function of a shallow source is much smaller than that of compressibility, while the effect of gravitation on the post-seismic Green's function over longer timescales is much greater than that of compressibility. Finally, taking the 2011 Tohoku-Oki Mw9.0 and 2013 Okhotsk Mw 8.3 earthquakes as examples, the simulation results show that post-seismic deformation studies should consider the Earth's compressibility, which is more obvious in the early post-seismic period. Likewise, when the viscosity of the study area is low, the effect of gravitation cannot be ignored. For the slip inversion of Tohoku-Oki Mw 9.0 earthquake, the incompressible Earth model leads to a smaller slip magnitude than the compressible Earth model.

Maximum mass of anisotropic charged strange quark stars in a higher dimensional approach (D ≥ 4)

In this article, a new class of solutions of Einstein-Maxwell field equations of relativistic strange quark stars obtained in dimensions , is shown. We assume that the geometry of space-time is pseudo-spheroid, embedded in Euclidean space of dimensions. The MIT bag model equation of state is employed to study the relevant properties of strange quark stars. For the causal and non-negative nature of the square of the radial sound velocity , we observe that some restrictions exist on the reduced radius , where R is a parameter related to the curvature of the space-time, and b is the radius of the star. The spheroidal parameter λ used here defines the metric potential of the component, which is pseudo-spheroidal in nature. We note that the pressure anisotropy and charge have some effects on λ. The maximum mass for a given surface density () or bag constant assumes a maximum value in dimension and decreases for other values of D. The generalized Buchdahl limit for a higher dimensional charged star is also obeyed in this model. We observe that in this model, we can predict the mass of a strange quark star using a suitable value of the electric charge (Q) and bag constant (B). Energy and stability conditions are also satisfied in this model. Stability is also studied considering the dependence of the Lagrangian perturbation of radial pressure () on the frequency of normal modes of oscillations. The tidal Love number and tidal de-formability are also evaluated.

Generalized Regge-Wheeler equation from Effective Field Theory of black hole perturbations with a timelike scalar profile

Recently, the Effective Field Theory (EFT) of perturbations on an arbitrary background metric with a timelike scalar profile was formulated in the context of scalar-tensor theories. Here, we generalize the dictionary between the coefficients in the EFT action and those in covariant theories to accommodate shift- and reflection-symmetric quadratic higher-order scalar-tensor theories, including DHOST as well as U-DHOST. We then use the EFT action to study the dynamics of odd-parity perturbations on a static and spherically symmetric black hole background with a timelike scalar profile. Finally, we obtain the generalized Regge-Wheeler equation that can be used, e.g., to determine the spectrum of quasinormal modes and tidal Love numbers.

Central pressure-dependent compact anisotropic stellar model and its tidal Love number

Central pressure-dependent compact anisotropic stellar model and its tidal Love number

Near-zone symmetries of Kerr black holes

We study the near-zone symmetries of a massless scalar field on four-dimensional black hole backgrounds. We provide a geometric understanding that unifies various recently discovered symmetries as part of an SO(4, 2) group. Of these, a subset are exact symmetries of the static sector and give rise to the ladder symmetries responsible for the vanishing of Love numbers. In the Kerr case, we compare different near-zone approximations in the literature, and focus on the implementation that retains the symmetries of the static limit. We also describe the relation to spin-1 and 2 perturbations.

Contribution of GRACE Satellite Mission to the Determination of Orthometric/Normal Heights Corrected for Their Dynamics—A Case Study of Poland

Physical heights were traditionally determined without considering the dynamic processes of the Earth induced from temporal mass variations. The Gravity Recovery and Climate Experiment (GRACE) mission provided valuable data that allow the estimation of geoid/quasigeoid height changes and vertical deformations of the Earth’s surface induced from temporal mass loading, and thereby temporal variations of physical heights. The objective of this investigation is to discuss the determination of orthometric/normal heights considering mass transports within the Earth’s system. An approach to determine such heights was proposed. First, temporal variations of orthometric/normal heights (ΔH/ΔH*) were determined using the release 6 GRACE-based Global Geopotential Models together with load Love numbers obtained from the preliminary reference Earth model. Then, those variations were modelled and predicted using the seasonal decomposition (SD) method. The proposed approach was tested on the territory of Poland. The main results obtained reveal that ΔH/ΔH* over the area investigated are at the level of a couple of centimetres and that they can be modelled and predicted with a millimetre accuracy using the SD method. Orthometric/normal heights corrected for their dynamics can be determined by combining modelled ΔH/ΔH* with orthometric/normal heights referred to a specific reference epoch.

Quadratic-in-spin interactions at fifth post-Newtonian order probe new physics

We obtain for the first time all quadratic-in-spin interactions in spinning binaries at the third subleading order in post-Newtonian (PN) gravity, and provide their observable binding energies and their gauge-invariant relations to the angular momentum. Our results are valid for generic compact objects, orbits, and spin orientations, and enter at the fifth PN order for maximally-rotating objects, thus pushing the state of the art. This is accomplished through an extension of the effective field theory of spinning gravitating objects, and of its computational application. We also discover a new finite-size effect which is unique to spinning objects, with a new “Spin Love number” as its characteristic coefficient, that is a new probe for gravity and QCD.

Callisto and Europa Gravity Measurements from JUICE 3GM Experiment Simulation

The JUpiter Icy Moons Explorer is an ESA mission set for launch in 2023 April and arrival in the Jovian system in 2031 July to investigate Jupiter and its icy satellites with a suite of 10 instruments. The mission will execute several flybys of the icy moons Europa, Callisto, and Ganymede before ending the mission with a 9-month orbit around Ganymede. The 3GM experiment on board the spacecraft will exploit accurate range and Doppler (range-rate) measurements to determine the moons’ orbit, gravity field, and tidal deformation. The focus of this paper is on the retrieval of Europa’s and Callisto’s gravity field, without delving into the modeling of their interior structures. By means of a covariance analysis of the data acquired during flybys, we assess the expected results from the 3GM gravity experiment. We find that the two Europa flybys will provide a determination of the J 2 and C 22 quadrupole gravity field coefficients with an accuracy of 3.8 × 10−6 and 5.1 × 10−7, respectively. The 21 Callisto flybys will provide a determination of the global gravity field to approximately degree and order 7, the moon ephemerides, and the time-variable component of the gravitational tide raised by Jupiter on the moon. The k 2 Love number, describing the Callisto tidal response at its orbital period, can be determined with an uncertainty σ k2 ∼ 0.06, allowing us to distinguish with good confidence between a moon with or without an internal ocean. The constraints derived by 3GM gravity measurements can then be used to develop interior models of the moon.

Tidal deformability of crystallized white dwarfs in full general relativity

Space-based gravitational-wave detectors offer new prospects for probing the interior of white dwarfs in binary systems through the imprints of tidal effects on the gravitational-wave signal. Some of the binaries that will be observed could have evolved for long enough for the white dwarfs to be at least partially crystallized. The apsidal motion constant ${k}_{2}$ (also called the second gravitoelectric Love number) of a cold crystallized white dwarf is computed in full general relativity considering different compositions. The elasticity of the crystallized core is found to systematically reduce the tidal deformability, especially for low-mass stars. Fully relativistic results are compared to those obtained in Newtonian gravity. It is shown that the relativistic correction to the observable tidal deformability ${k}_{2}{R}^{5}$ (where $R$ is the stellar radius) is negligible for low-mass white dwarfs but becomes increasingly important for more massive white dwarfs. When approaching the maximum mass, the application of Newtonian theory instead of general relativity leads to dramatic errors. The case of eccentric binaries, for which the precession of the periastron causes a frequency splitting of the gravitational-wave signal depending on the apsidal motion constants of the two stars, is investigated. Future measurements of the precession rate by the Laser Interferometer Space Antenna, which is planned to be in operation within the next decade, could potentially provide estimates of the individual masses. It is found that the errors incurred by the neglect of the elasticity of the crystallized core could be very large, especially for low-mass white dwarfs. Gravitational-wave observations could thus provide a new way to study the crystallization of white dwarfs.

Impact and detectability of spin-tidal couplings in neutron star inspirals

The gravitational wave signal from a binary neutron star merger carries the imprint of the deformability properties of the coalescing bodies, and then of the equation of state of neutron stars. In current models of the waveforms emitted in these events, the contribution of tidal deformation is encoded in a set of parameters, the tidal Love numbers. More refined models include tidal-rotation couplings, described by an additional set of parameters, the rotational tidal Love numbers, which appear in the waveform at $6.5$ post-Newtonian order. For neutron stars with spins as large as $\sim0.1$, we show that neglecting tidal-rotation couplings may lead to a significant error in the parameter estimation by third-generation gravitational wave detectors. By performing a Fisher matrix analysis we assess the measurability of rotational tidal Love numbers, showing that their contribution in the waveform could be measured by third-generation detectors. Our results suggest that current models of tidal deformation in late inspiral should be improved in order to avoid waveform systematics and extract reliable information from gravitational wave signals observed by next generation detectors.

On computing viscoelastic Love numbers for general planetary models: the ALMA3 code

SUMMARY The computation of the Love numbers (LNs) for a spherically symmetric self-gravitating viscoelastic Earth is a classical problem in global geodynamics. Here we revisit the problem of the numerical evaluation of loading and tidal LNs in the static limit for an incompressible planetary body, adopting a Laplace inversion scheme based upon the Post-Widder formula as an alternative to the traditional viscoelastic normal modes method. We also consider, within the same framework, complex-valued, frequency-dependent LNs that describe the response to a periodic forcing, which are paramount in the study of the tidal deformation of planets. Furthermore, we numerically obtain the time-derivatives of LNs, suitable for modelling geodetic signals in response to surface loads variations. A number of examples are shown, in which time and frequency-dependent LNs are evaluated for the Earth and planets adopting realistic rheological profiles. The numerical solution scheme is implemented in ALMA3 (the plAnetary Love nuMbers cAlculator, version 3), an upgraded open-source Fortran 90 program that computes the LNs for radially layered planetary bodies with a wide range of rheologies, including transient laws like Andrade or Burgers.

Hidden symmetry of the static response of black holes: applications to Love numbers

We show that any static linear perturbations around Schwarzschild black holes enjoy a set of conserved charges which forms a centrally extended Schrödinger algebra mathfrak{sh} (1) = mathfrak{sl} (2, ℝ) ⋉ mathcal{H} . The central charge is given by the black hole mass, echoing results on black hole entropy from near-horizon diffeomorphism symmetry. The finite symmetry transformations generated by these conserved charges correspond to Galilean and conformal transformations of the static field and of the coordinates. This new structure allows one to discuss the static response of a Schwarzschild black hole in the test field approximation from a symmetry-based approach. First we show that the (horizontal) symmetry protecting the vanishing of the Love numbers recently exhibited by Hui et al., dubbed the HJPSS symmetry, coincides with one of the mathfrak{sl} (2, ℝ) generators of the Schrödinger group. Then, it is demonstrated that the HJPSS symmetry is selected thanks to the spontaneous breaking of the full Schrödinger symmetry at the horizon down to a simple abelian sub-group. The latter can be understood as the symmetry protecting the regularity of the test field at the horizon. In the 4-dimensional case, this provides a symmetry protection for the vanishing of the Schwarzschild Love numbers. Our results trivially extend to the Kerr case.

Constraining the Internal Structures of Venus and Mars from the Gravity Response to Atmospheric Loading

The gravity fields of celestial bodies that possess an atmosphere are periodically perturbed by the redistribution of fluid mass associated with atmospheric dynamics. A component of this perturbation is due to the gravitational response of the body to the deformation of its surface induced by the atmospheric pressure loading. The magnitude of this effect depends on the relation between the loading and the response in terms of geopotential variations measured by the load Love numbers. In this work, we simulate and analyze the gravity field generated by the atmospheres of Venus and Mars by accounting for different models of their internal structure. By precisely characterizing the phenomena that drive the mass transportation in the atmosphere through general circulation models, we determine the effect of the interior structure on the response to the atmospheric loading. An accurate estimation of the time-varying gravity field, which measures the atmospheric contribution, may provide significant constraints on the interior structure through the measurement of the load Love numbers. A combined determination of tidal and load Love numbers would enhance our knowledge of the interior of planetary bodies, providing further geophysical constraints in the inversion of internal structure models.

Possible tidal dissipation in millisecond pulsar binaries

Aims. The post-Keplerian(PK) parameters inferred from pulsar timing provide a convenient way to test Einstein’s general theory of relativity. However, before obtaining a pure orbital decay Ṗb induced by gravitational wave radiation, which is one of the PK parameters, a number of factors need to be accounted for carefully. The effect of tidal dissipation on Ṗb has been thought of as negligible. Here, we investigate the data for possible effects of tidal dissipation on Ṗb. Methods. The possibility of the tidal dissipation as a contributor to Ṗb in a large sample of millisecond pulsar binaries is investigated in detail. Results. We collected a large sample of pulsar binaries with measured Ṗb. All of the systems are millisecond pulsars. The residual ṖbRes of these systems was obtained by subtracting the three normal effects, that is to say the effect of Shklovskii, line-of-sight acceleration, and gravitational radiation. Assuming that tidal dissipation is responsible for such a residual ṖbResa, the tidal parameters of these systems can be calculated and compared with the tidal models. Conclusions. The residual ṖbResa is distributed over the half positive and half negative. The dynamical tidal model can explain the residual Ṗb of millisecond pulsar-white dwarf binaries. And the Love number of the main-sequence companion of PSR J1227-4853 can be derived as a reasonable value k2 = 0.177−0.058+0.098 with the equilibrium tidal model. Those results are compatible with the scenario of tidal dissipation. Additionally, a weak correlation between the tidal parameter and orbital period is revealed, likely originating in the tidal process of the recycled stage which is worthy of further investigation.

Frequency-dependent Ganymede’s tidal Love number [formula omitted] detection by JUICE’s 3GM experiment and implications for the subsurface ocean thickness

Ganymede will be the first icy satellite in the Solar System orbited by a spacecraft, ESA’s JUpiter ICy moons Explorer (JUICE). JUICE launch is scheduled for April 2023 and the arrival at Ganymede is foreseen in 2035.Thanks to the advanced Ka-band radio tracking system, the Geodesy and Geophysics of Jupiter and Galilean Moons (3GM) experiment aboard the mission will provide range and range-rate measurements that will be used to infer the static (up to degree 35–45) and time-varying gravity field and the internal structure of the moon. Ganymede is subject to tidal deformation, which periodically modifies its gravity field. The larger contribution to the time-varying gravity is due to the tidal interaction with Jupiter, modulated by the eccentricity and the inclination of the moon’s orbit. In addition, Ganymede also experiences a lower amplitude time-varying tidal interaction with Io, Europa and Callisto. To a good approximation, the corresponding gravitational signals are periodic and they contain several harmonics of the fundamental synodic frequencies. The elastic response of Ganymede is expected to be frequency-dependent as well. In this work we describe a procedure to estimate the Ganymede’s tidal Love number k2 at different frequencies in the orbit determination process of the 3GM experiment, supported by numerical simulations. Finally, we show that 3GM measurements can be used to characterize the interior structure of Ganymede, providing a tight constraint on the subsurface ocean thickness. The ocean density can be also constrained, but with a lower precision.