Love Number Research Articles

Mercury's Tidal Love Number h2 ${h}_{2}$ From Co‐Registration of MLA Profiles

AbstractDue to its eccentric orbit, Mercury experiences a varying gravitational pull from the Sun along its orbit, leading to periodic surface tidal deformation. The previous measurement of Mercury's tidal by Bertone et al. (2021, https://doi.org/10.1029/2020je006683) is based on minimizing height differences at cross‐overs of the Mercury Laser Altimeter (MLA) profiles. However, this method can suffer from significant interpolation errors. In this study, we apply an alternative approach, which is based on the co‐registration of reprocessed MLA profiles. For the reprocessing, we account for the pointing aberration and incorporate an updated spacecraft orbit model. Within the study region of 77°N to 84°N, we obtain a tidal of 0.920.58 (3‐). This value is compatible with current interior structure and rheology models, but significantly lower than the previous estimate of 1.550.65 (3‐). When combined with recent tidal estimates, our measurement favors a small to medium‐sized inner core.

18.6-year tidal variations observed from 48-year Satellite Laser Ranging (SLR)

Summary Analysis of the 48-year of satellite laser ranging (SLR) data of multiple satellites shows that long-term variations in the Earth's dynamical oblateness represented by the second-degree zonal harmonic J2 is best characterized by the superposition of a quadratic trend, 10.5-year and 18.6-year variations. These variations result from climate-related mass changes, tides, and core flow-induced variations at the core-mantle boundary (CMB). We determined that the global ocean's response to the lunar attraction at the 18.6-year period is near equilibrium, with an amplitude of 0.4735 ± 0.008 cm and an error of ∼11% relative to the modeled amplitude (0.4224 cm), and ∼2±3 degrees of phase lag. The 18.6-year Love Number was found to be 0.01375-i0.00553 with an error of 2% for both the real and imaginary parts and a phase correction π for the imaginary part of the IERS 2010 anelasticity model. The nominal frequency-independent anelasticity Love number, k₂, was determined to be 0.3022 ± 0.0001 for the 18.6-year period, based on a reference frequency of 200 seconds and α = 0.1514 for mantle anelasticity for mantle anelasticity. This study also reveals a significant gravitational signal (3.06×10−11) in J2 obstructs the Earth's mantle anelastic response to the 18.6-year tidal forces, reflecting in phase shift of π the imaginary part of the IERS 2010 Love number. This signal can be characterized by a positive Love number of 0.01106 in the modeling of the variation in J2 coupling with the 18.6-year tide. This signal is possibly produced by the core dynamics, which creates a gravitational signal in J2 with an amplitude of 3.36×10−11 at the decadal time scale and could account for ∼70% of the observed 10.5-year variation.

Can Neutron Star Tidal Effects Obscure Deviations from General Relativity?

Abstract One of the main goals of gravitational-wave astrophysics is to study gravity in the strong-field regime and constrain deviations from general relativity (GR). Any such deviation affects not only binary dynamics and gravitational-wave emission but also the structure and tidal properties of compact objects. In the case of neutron stars, masses, radii, and tidal deformabilities can all differ significantly between different theories of gravity. Currently, the measurement uncertainties in neutron star radii and tidal deformabilities are quite large. However, much less is known about how the large uncertainty in the nuclear equation of state (EOS) might affect tests of GR using binary neutron star mergers. Conversely, using the wrong theory of gravity might lead to incorrect constraints on the nuclear EOS. Here, we study this problem within scalar–tensor (ST) theory. We apply the recently derived ℓ = 2 tidal Love numbers in this theory to parameter estimation of GW170817. Correspondingly, we test if physics beyond GR could bias measurements of the nuclear EOS and neutron star radii. We find that parameter inference for both the GR and ST cases returns consistent component masses and tidal deformabilities. The radius and the EOS posteriors, however, differ between the two theories, but neither is excluded by current observational limits. This indicates that measurements of the nuclear EOS may be biased and that deviations from GR could go undetected when analyzing current binary neutron star mergers.

Symmetries of vanishing nonlinear Love numbers of Schwarzschild black holes

The tidal Love numbers parametrize the conservative induced tidal response of self-gravitating objects. It is well established that asymptotically-flat black holes in four-dimensional general relativity have vanishing Love numbers. In linear perturbation theory, this result was shown to be a consequence of ladder symmetries acting on black hole perturbations. In this work, we show that a black hole’s tidal response induced by a static, parity-even tidal field vanishes for all multipoles to all orders in perturbation theory. Our strategy is to focus on static and axisymmetric spacetimes for which the dimensional reduction to the fully nonlinear Weyl solution is well-known. We define the nonlinear Love numbers using the point-particle effective field theory, matching with the Weyl solution to show that an infinite subset of the static, parity-even Love number couplings vanish, to all orders in perturbation theory. This conclusion holds even if the tidal field deviates from axisymmetry. Lastly, we discuss the symmetries underlying the vanishing of the nonlinear Love numbers. An (2, ℝ) algebra acting on a covariantly-defined potential furnishes ladder symmetries analogous to those in linear theory. This is because the dynamics of the potential are isomorphic to those of a static, massless scalar on a Schwarzschild background. We comment on the connection between the ladder symmetries and the Geroch group that is well-known to arise from dimensional reduction.

Lunar Degree-2 Tidal Love Number Determination Based on Combination of Four-way Radiometric Tracking and LLR Data

Abstract Lunar tidal Love numbers ( k 2 , h 2 , and l 2 ), critical for understanding lunar interior structures, are traditionally derived from methods such as lunar spacecraft radiometric tracking, Lunar Laser Ranging (LLR), and laser altimetry. The Chang’e 7 lander is expected to carry a radio transponder and a retroreflector. In order to assess the ability of the Chang’e 7 data to constrain these Love numbers, we conducted a numerical simulation based on a four-way relay tracking model in terms of different orbital inclinations, varying lander positions (nearside, south pole, and farside of the Moon), combined with LLR. Our simulation results demonstrate the effectiveness of the Chang’e 7 in determining tidal Love numbers, with a particular advantage for h 2 and l 2 . By combining the LLR data from the nearside and the south pole of the Moon as well as the four-way lander tracking data of Chang’e 7, the formal uncertainties of h 2 and l 2 are better than 10 − 4 , which is nearly 1 order of magnitude higher than the existing accuracy estimated by LLR alone ( 10 − 3 ). For the lunar south pole region, four-way radiometric ranging can still offer higher sensitivity and a longer observation time compared to LLR. This is evident for measurements of h 2 , despite the fact that radiometric ranging precision is an order of magnitude lower than that of LLR. This type of observation not only complements laser altimetry and LLR but also holds potential for application to other celestial bodies (e.g., Ganymede, Callisto, Mars) capable of supporting a landed mission.

The vanishing of the non-linear static love number of Kerr black holes and the role of symmetries

We investigate the tidal response of Kerr black holes in four-dimensional spacetimes subjected to external gravitational fields. Using the Ernst formalism and Weyl coordinates, we analyze the non-linear tidal deformation of rotating black holes and demonstrate that their static tidal Love numbers vanish at all orders of the external tidal field. We also show that this result is intimately related to the presence of underlying non-linear symmetries. Our analysis generalizes previous findings for Schwarzschild black holes and confirms the robustness of four-dimensional black holes against tidal forces.

Vanishing of quadratic Love numbers of Schwarzschild black holes

The induced conservative tidal response of self-gravitating objects in general relativity is parametrized in terms of a set of coefficients, which are commonly referred to as Love numbers. For asymptotically-flat black holes in four spacetime dimensions, the Love numbers are famously zero in the static regime. In this work, we show that this result continues to hold upon inclusion of nonlinearities in the theory for Schwarzschild black holes. We first solve the quadratic Einstein equations in the static limit to all orders in the multipolar expansion, including both even and odd perturbations. We show that the second-order solutions take simple analytic expressions, generically expressible in the form of finite polynomials. We then define the quadratic Love numbers at the level of the point-particle effective field theory. By performing the matching with the full solution in general relativity, we show that quadratic Love number coefficients are zero to all orders in the derivative expansion, like the linear ones.

Dark I-Love-Q

For neutron stars, there exist universal relations insensitive to the equation of states, the so called I-Love-Q relations, which show the connections among the moment of inertia, tidal Love number and quadrupole moment. In this paper, we show that these relations also apply to dark stars, bosonic or fermionic. The relations can be extended to higher ranges of the variables, clarifying the deviations for dark stars in the literature, as those curves all approximate the ones generated by a polytropic equation of state, when taking the low density (pressure) limit. Besides, we find that for equation of states with scaling symmetries, the I-Love-Q curves do not change when adjusting the scaling parameters.

Charging the Love numbers: Charged scalar response coefficients of Kerr-Newman black holes

Charging the Love numbers: Charged scalar response coefficients of Kerr-Newman black holes

Characterising WASP-43b’s interior structure: Unveiling tidal decay and apsidal motion

Context. Recent developments in exoplanetary research highlight the importance of Love numbers in understanding the internal dynamics, formation, migration history, and potential habitability of exoplanets. Love numbers represent crucial parameters that gauge how exoplanets respond to external forces such as tidal interactions and rotational effects. By measuring these responses, insights into the internal structure, composition, and density distribution of exoplanets can be gained. The rate of apsidal precession of a planetary orbit is directly linked to the second-order fluid Love numbers. Thus, Love numbers can also offer valuable insights into the mass distribution of a planet. Aims. In this context, we aim to re-determine the orbital parameters of WASP-43b – in particular, the orbital period, eccentricity, and argument of the periastron – and its orbital evolution. We study the outcomes of the tidal interaction with the host star in order to identify whether tidal decay and periastron precession occur in the system. Methods. We observed WASP-43b with HARPS, whose data we present for the first time, and we also analysed the newly acquired JWST full-phase light curve. We jointly fit new and archival radial velocity and transit and occultation mid-times, including tidal decay, periastron precession, and long-term acceleration in the system. Results. We detected a tidal decay rate of Ṗa = (−l.99±0.50) ms yr−1 and a periastron precession rate of ω = 0.1727−0.0089+0.0083)° d−1 = (621.72−32.04+29.88)″ d−1). This is the first time that both periastron precession and tidal decay are simultaneously detected in an exoplanetary system. The observed tidal interactions can neither be explained by the tidal contribution to apsidal motion of a non-aligned stellar or planetary rotation axis nor by assuming a non-synchronous rotation for the planet, and a value for the planetary Love number cannot be derived. Moreover, we excluded the presence of a second body (e.g. a distant companion star or a yet undiscovered planet) down to a planetary mass of ≳0.3 MJ and up to an orbital period of ≲3700 days. We leave the question of the cause of the observed apsidal motion open.

Anisotropic tidal dissipation in misaligned planetary systems

Context. Tides are the main driving force behind the long-term evolution of planetary systems. The associated energy dissipation and momentum exchanges are commonly described by Love numbers, which relate the exciting potential to the tidally perturbed potential. These transfer functions are generally assumed to depend solely on tidal frequency and body rheology, following the isotropic assumption, which presumes invariance of properties by rotation about the centre of mass. Aims. We examine the limitations of the isotropic assumption for fluid bodies, in which Coriolis acceleration breaks spherical symmetry, resulting in rotational scattering and complex tidal responses. Methods. Using angular momentum theory, we derived a new formalism to calculate the tidal rates of energy and momentum transfers in non-isotropic cases. We applied this formalism to the Earth-Moon system to assess the effects of anisotropy in planet-satellite systems with misaligned spin and orbital angular momenta. Results. Our findings indicate that the isotropic assumption can introduce significant errors in planetary evolution models, particularly in the dynamical tide regime. These errors stem from forced wave resonances, with inaccuracies in energy dissipation scaling in proportion to resonance amplification factors.

Constraining the parameters of the Andrade rheology in Earth's mantle with Love numbers of 12 tidal constituents

Constraining the parameters of the Andrade rheology in Earth's mantle with Love numbers of 12 tidal constituents

Tidal Love numbers of static black holes in anti-de Sitter

Tidal Love numbers of anti-de Sitter black holes are understood as linear response coefficients governing how the holographically dual plasma polarizes when the geometry of the space, in which the plasma lives, is deformed. So far, this picture has been applied only to black branes with plane wave perturbations. We fill the gap in the literature by performing the computation of tidal Love numbers for the four-dimensional Schwarzschild solution in global anti-de Sitter, which is dual to a conformal plasma on S2. We conclude about the effect of the bulk gravitational perturbations on the boundary metric and stress tensor, responsible for the geometric polarization. The computation of the tidal Love numbers is performed in both Regge-Wheeler gauge and the Kodama-Ishibashi gauge-invariant approach. We spell out how to convert the tidal Love numbers determined in these two formalisms and find perfect agreement. We also relate the Kodama-Ishibashi formalism with the Kovtun-Starinets approach, which is particularly well suited for the holographic analysis of black branes. This allows us to compare with the tidal Love number results for black branes in anti-de Sitter, also finding agreement in the relevant regime.

Io’s tidal response precludes a shallow magma ocean

Io experiences tidal deformation as a result of its eccentric orbit around Jupiter, which provides a primary energy source for Io’s continuing volcanic activity and infrared emission1. The amount of tidal energy dissipated within Io is enormous and has been suggested to support the large-scale melting of its interior and the formation of a global subsurface magma ocean. If Io has a shallow global magma ocean, its tidal deformation would be much larger than in the case of a more rigid, mostly solid interior2. Here we report the measurement of Io’s tidal deformation, quantified by the gravitational tidal Love number k2, enabled by two recent flybys of the Juno spacecraft. By combining Juno3,4 and Galileo5, 6–7 Doppler data from the NASA Deep Space Network and astrometric observations, we recover Re(k2) of 0.125 ± 0.047 (1σ) and the tidal dissipation parameter Q of 11.4 ± 3.6 (1σ). These measurements confirm that a shallow global magma ocean in Io does not exist and are consistent with Io having a mostly solid mantle2. Our results indicate that tidal forces do not universally create global magma oceans, which may be prevented from forming owing to rapid melt ascent, intrusion and eruption8,9, so even strong tidal heating—such as that expected on several known exoplanets and super-Earths10—may not guarantee the formation of magma oceans on moons or planetary bodies.

CHEOPS observations confirm nodal precession in the WASP-33 system

We aim to observe the transits and occultations of which orbits a rapidly rotating delta Scuti pulsator, with the goal of measuring the orbital obliquity via the gravity-darkening effect, and constraining the geometric albedo via the occultation depth. We observed four transits and four occultations with CHEOPS, and employ a variety of techniques to remove the effects of the stellar pulsations from the light curves, as well as the usual CHEOPS systematic effects. We also performed a comprehensive analysis of low-resolution spectral and Gaia data to re-determine the stellar properties of system. We measure an orbital obliquity degrees, which is consistent with previous measurements made via Doppler tomography. We also measure the planetary impact parameter, and confirm that this parameter is undergoing rapid secular evolution as a result of nodal precession of the planetary orbit. This precession allows us to determine the second-order fluid Love number of the star, which we find agrees well with the predictions of theoretical stellar models. We are unable to robustly measure a unique value of the occultation depth, and emphasise the need for long-baseline observations to better measure the pulsation periods.

Determination of the Andrade Rheological Model Parameters for the Earth’s Mantle from the Love Numbers of Ten Tidal Components

Determination of the Andrade Rheological Model Parameters for the Earth’s Mantle from the Love Numbers of Ten Tidal Components

The Load Love Numbers for Various Rheological Models of Venus

The Load Love Numbers for Various Rheological Models of Venus

Scalar tidal response of a rotating BTZ black hole

We study the response of a rotating BTZ black hole to the scalar tidal perturbation. We show that the real component of the tidal response function isn’t zero, indicating that a rotating BTZ black hole possesses non-zero tidal Love numbers. Additionally, we observe scale-dependent behaviour, known as log-running, in the tidal response function. We also conduct a separate analysis on an extremal rotating BTZ black hole, finding qualitative similarities with its non-extremal counterpart. In addition, we present a procedure to calculate the tidal response function of a charged rotating BTZ black hole as well.

Tidal Love numbers of anisotropic stars within the complexity factor formalism

Tidal Love numbers of anisotropic stars within the complexity factor formalism

Investigating tidal heating in neutron stars via gravitational Raman scattering

We present a scattering amplitude formalism to study the tidal heating effects of nonspinning neutron stars incorporating both worldline effective field theory and relativistic stellar perturbation theory. In neutron stars, tidal heating arises from fluid viscosity due to various scattering processes in the interior. It also serves as a channel for the exchange of energy and angular momentum between the neutron star and its environment. In the interior of the neutron star, we first derive two master perturbation equations that capture fluid perturbations accurate to linear order in frequency. Remarkably, these equations receive no contribution from bulk viscosity due to a peculiar adiabatic incompressibility which arises in stellar fluid for nonbarotropic perturbations. In the exterior, the metric perturbations reduce to the Regge-Wheeler equation which we solve using the analytical Mano-Suzuki-Takasugi method. We compute the amplitude for gravitational waves scattering off a neutron star, also known as gravitational Raman scattering. From the amplitude, we obtain expressions for the electric quadrupolar static Love number and the leading dissipation number to all orders in compactness. We then compute the leading dissipation number for various realistic equations of state and estimate the change in the number of gravitational-wave cycles due to tidal heating during inspiral in the LIGO-Virgo-KAGRA band. Published by the American Physical Society 2024