Tidal Love Number Research Articles

The tides of Mercury and possible implications for its interior structure

The combination of the radio tracking of the MErcury Surface, Space ENvironment, GEochemistry, and Ranging spacecraft and Earth‐based radar measurements of the planet's spin state gives three fundamental quantities for the determination of the interior structure of Mercury: mean density ρ, moment of inertia C, and moment of inertia of the outer solid shell Cm. This work focuses on the additional information that can be gained by a determination of the change in gravitational potential due to planetary tides, as parameterized by the tidal potential Love number k2. We investigate the tidal response for sets of interior models that are compatible with the available constraints (ρ, C, and Cm). We show that the tidal response correlates with the size of the liquid core and the mean density of material below the outer solid shell and that it is affected by the rheology of the outer solid shell of the planet, which depends on its temperature and mineralogy. For a mantle grain size of 1 cm, we calculate that the tidal k2 of Mercury is in the range 0.45 to 0.52. Some of the current models for the interior structure of Mercury are compatible with the existence of a solid FeS layer at the top of the core. Such a layer, if present, would increase the tidal response of the planet.

Effectively universal behavior of rotating neutron stars in general relativity makes them even simpler than their Newtonian counterparts.

Recently, it was shown that slowly rotating neutron stars exhibit an interesting correlation between their moment of inertia I, their quadrupole moment Q, and their tidal deformation Love number λ (the I-Love-Q relations), independently of the equation of state of the compact object. In the present Letter a similar, more general, universality is shown to hold true for all rotating neutron stars within general relativity; the first four multipole moments of the neutron star are related in a way independent of the nuclear matter equation of state we assume. By exploiting this relation, we can describe quite accurately the geometry around a neutron star with fewer parameters, even if we don't know precisely the equation of state. Furthermore, this universal behavior displayed by neutron stars could promote them to a more promising class of candidates (next to black holes) for testing theories of gravity.

Shape, topography, gravity anomalies and tidal deformation of Titan

Gravity measurements and elevation data from the Cassini mission have been used to create shape, global topography and gravity anomaly models of Titan that enable an improved understanding of its outer ice I shell structure. We provide constraints on the averaged ice shell thickness and its long-wavelength lateral variations, as well as the density of the subsurface ocean using gravity anomalies, the tidal Love number k2 measurement and long-wavelength topography. We found that Titan’s surface topography is consistent with an approximate isostatically compensated ice shell of variable thickness, likely in a thermally conductive or in a subcritical convective state, overlying a relatively dense subsurface ocean.

Viscoelastic tidal dissipation in giant planets and formation of hot Jupiters through high-eccentricity migration

We study the possibility of tidal dissipation in the solid cores of giant planets and its implication for the formation of hot Jupiters through high-eccentricity migration. We present a general framework by which the tidal evolution of planetary systems can be computed for any form of tidal dissipation, characterized by the imaginary part of the complex tidal Love number, ${\rm Im}[{\tilde k}_2(\omega)]$, as a function of the forcing frequency $\omega$. Using the simplest viscoelastic dissipation model (the Maxwell model) for the rocky core and including the effect of a nondissipative fluid envelope, we show that with reasonable (but uncertain) physical parameters for the core (size, viscosity and shear modulus), tidal dissipation in the core can accommodate the tidal-Q constraint of the Solar system gas giants and at the same time allows exoplanetary hot Jupiters to form via tidal circularization in the high-e migration scenario. By contrast, the often-used weak friction theory of equilibrium tide would lead to a discrepancy between the Solar system constraint and the amount of dissipation necessary for high-e migration. We also show that tidal heating in the rocky core can lead to modest radius inflation of the planets, particularly when the planets are in the high-eccentricity phase ($e\sim 0.6$) during their high-e migration. Finally, as an interesting by-product of our study, we note that for a generic tidal response function ${\rm Im}[{\tilde k}_2(\omega)]$, it is possible that spin equilibrium (zero torque) can be achieved for multiple spin frequencies (at a given $e$), and the actual pseudo-synchronized spin rate depends on the evolutionary history of the system.

Dissipation at tidal and seismic frequencies in a melt‐free, anhydrous Mars

The measured inward motion of Phobos provides a constraint on the tidal dissipation factor, Q, within Mars. We model viscoelastic dissipation inside a convective Mars using a modified Burgers model based on laboratory experiments on anhydrous, melt‐free olivine. The model tidal Q is highly sensitive to the mantle potential temperature and grain size assumed but relatively insensitive to the bulk density and rigidity structure. Q thus provides a tight constraint on the Martian interior temperature. By fitting the observed tidal Q and tidal Love number (k2) values and requiring present‐day melt generation, we estimate that for a grain size of 1 cm the current mantle potential temperature is 1625±75 K, similar to that of the Earth. This estimate is consistent with recent petrologically derived determinations of mantle potential temperature but lower than estimates in some thermal evolution models. The presence of water in the Martian mantle would reduce our estimated temperature. Our preferred mantle grain size of ≈1 cm is somewhat larger than that of the Earth's upper mantle. The predicted mantle seismic Q is about 130 and is almost independent of depth. The Martian lithosphere represents a high seismic velocity lid, which should be readily detectable with future seismological observations.

Advanced LIGO’s ability to detect apparent violations of the cosmic censorship conjecture and the no-hair theorem through compact binary coalescence detections

We study the ability of the advanced Laser Interferometer Gravitational-wave Observatory (aLIGO) to detect apparent violations of the cosmic censorship conjecture and the no-hair theorem. The cosmic censorship conjecture, which is believed to be true in the theory of general relativity, limits the spin-to-mass-squared ratio of a Kerr black hole. The no-hair theorem, which is also believed to be true in the theory of general relativity, suggests a particular value for the tidal Love number of a non-rotating black hole. Using the Fisher matrix formalism, we examine the measurability of the spin and tidal deformability of compact binary systems involving at least one putative black hole. Using parameter measurement errors and correlations obtained from the Fisher matrix, we determine the smallest detectable violation of bounds implied by the cosmic censorship conjecture and the no-hair theorem. We examine the effect of excluding unphysical areas of parameter space when determining the smallest detectable apparent violations, and we examine the effect of different post-Newtonian corrections to the amplitude of the compact binary coalescence gravitational waveform. In addition, we perform a brief study of how the recently calculated 3.0 pN and 3.5 pN spin-orbit corrections to the phase affect spin and mass parameter measurability. We find that physical priors on the symmetric mass ratio and higher harmonics in the gravitational waveform could significantly affect the ability of aLIGO to investigate cosmic censorship and the no-hair theorem for certain systems.

ARE THEKEPLERNEAR-RESONANCE PLANET PAIRS DUE TO TIDAL DISSIPATION?

The multiple-planet systems discovered by the Kepler mission show an excess of planet pairs with period ratios just wide of exact commensurability for first-order resonances like 2:1 and 3:2. In principle, these planet pairs could have both resonance angles associated with the resonance librating if the orbital eccentricities are sufficiently small, because the width of first-order resonances diverges in the limit of vanishingly small eccentricity. We consider a widely-held scenario in which pairs of planets were captured into first-order resonances by migration due to planet-disk interactions, and subsequently became detached from the resonances, due to tidal dissipation in the planets. In the context of this scenario, we find a constraint on the ratio of the planet's tidal dissipation function and Love number that implies that some of the Kepler planets are likely solid. However, tides are not strong enough to move many of the planet pairs to the observed separations, suggesting that additional dissipative processes are at play.

I-Love-Q: Unexpected Universal Relations for Neutron Stars and Quark Stars

Neutron stars and quark stars are not only characterized by their mass and radius but also by how fast they spin, through their moment of inertia, and how much they can be deformed, through their Love number and quadrupole moment. These depend sensitively on the star's internal structure and thus on unknown nuclear physics. We find universal relations between the moment of inertia, the Love number, and the quadrupole moment that are independent of the neutron and quark star's internal structure. These can be used to learn about neutron star deformability through observations of the moment of inertia, break degeneracies in gravitational wave detection to measure spin in binary inspirals, distinguish neutron stars from quark stars, and test general relativity in a nuclear structure-independent fashion.

Tidal Love and Shida numbers estimated by geodetic VLBI

Frequency-dependent Love and Shida numbers, which characterize the Earth response to the tidal forces, were estimated in a global adjustment of all suitable geodetic Very Long Baseline Interferometry (VLBI) sessions from 1984.0 to 2011.0. Several solutions were carried out to determine the Love and Shida numbers for the tidal constituents at periods in the diurnal band and in the long-period band in addition to values of the Love and Shida numbers common for all tides of degree two. Adding up all twelve diurnal tidal waves that were estimated, the total differences in displacement with respect to the theoretical conventional values of the Love and Shida numbers calculated from an Earth model reach 1.73±0.29mm in radial direction and 1.15±0.15mm in the transverse plane. The difference in the radial deformation following from the estimates of the zonal Love numbers is largest for the semi-annual tide Ssa with 1.07±0.19mm.

Relation of dislocation Love numbers and conventional Love numbers and corresponding Green's functions for a surface rupture in a spherical earth model

SUMMARY This study was undertaken to complete the dislocation theory presented by Sun et al. for a spherical earth model with special treatment of coseismic deformations for a surface rupture. The previous theory is valid only for inner sources, but the coseismic deformations for a surface rupture source remain unsolved. Therefore, this study deals with the special case of coseismic deformations for a surface rupture dislocation. For this purpose, we first derive the relation between the dislocation Love numbers and the conventional Love numbers, that is, the tidal, press, and shear Love numbers, based on the Okubo reciprocity theorem. Then we derive the corresponding Green’s functions for coseismic displacement, potential/geoid, gravity and strain changes using Okubo asymptotic solutions. Finally, we present numerical results of those Green’s functions for four independent seismic sources. Then we discuss the computation of coseismic deformations at the singular point—epicentre. As a case study, the computations of the coseismic deformations caused by the 2011 Tohoku-Oki earthquake (Mw 9.0) with considering the new Green’s functions as a constrain validates the theoretical model established in this study.

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.

Evidence for Solid Planets from Kepler's Near-Resonance Systems

AbstractThe multiple-planet systems discovered by the Kepler mission show an excess of planet pairs with period ratios just wide of exact commensurability for first-order resonances like 2:1 and 3:2. In principle, these planet pairs could be in resonance if their orbital eccentricities are sufficiently small, because the width of first-order resonances diverges in the limit of vanishingly small eccentricity. We consider a widely-held scenario in which pairs of planets were captured into first-order resonances by migration due to planet-disk interactions, and subsequently became detached from the resonances, due to tidal dissipation in the planets. In the context of this scenario, we find a constraint on the ratio of the planet's tidal dissipation function and Love number that implies that some of the Kepler planets are likely solid. However, tides are not strong enough to move many of the planet pairs to the observed separations, suggesting that additional processes are at play.

ON THE DYNAMICS AND TIDAL DISSIPATION RATE OF THE WHITE DWARF IN 4U 1820-30

It has been suggested that the 170 day period in the light curve of the low mass X-ray binary 4U 1820-30 arises from the presence of a third body with a large inclination to the binary orbit. We show that this long period motion arises if the system is librating around the stable fixed point in a Kozai resonance. We demonstrate that mass transfer drives the system toward this fixed point, and calculate, both analytically and via numerical integrations, that the period of libration is of order 170 days when the mutual inclination is near the Kozai critical value. The non-zero eccentricity of the binary, combined with tidal dissipation, implies that the rate of change of the binary period would be slower than, or even of opposite sign to, that implied by standard mass transfer models. If the 170 day period results from libration, then, contrary to appearances, the orbital period of the inner binary is increasing with time; in that case, (e/0.009)^2Q/k_2 > 2.5 x 10^9, where k_2 = 0.01 is the tidal Love number and e = 0.009 is the fiducial eccentricity of the inner binary. It appears unlikely that the observed negative period derivative results from the smaller than expected (but positive) value of \dot P combined with the previously suggested acceleration of the system in the gravitational field of the host globular cluster NGC 6624. The discrepancy between the observed and expected period derivative requires further investigation.

Constraining the interior of extrasolar giant planets with the tidal Love numberk2using the example of HAT-P-13b

Transit and radial velocity observations continuously discover an increasing number of exoplanets. However, when it comes to the composition of the observed planets the data are compatible with several interior structure models. Thus, a planetary parameter sensitive to the planet's density distribution could help constrain this large number of possible models even further. We aim to investigate to what extent an exoplanet's interior can be constrained in terms of core mass and envelope metallicity by taking the tidal Love number k_2 into account as an additional possibly observable parameter. Because it is the only planet with an observationally determined k_2, we constructed interior models for the Hot Jupiter exoplanet HAT-P-13b by solving the equations of hydrostatic equilibrium and mass conservation for different boundary conditions. In particular, we varied the surface temperature and the outer temperature profile, as well as the envelope metallicity within the widest possible parameter range. We also considered atmospheric conditions that are consistent with nongray atmosphere models. For all these models we calculated the Love number k_2 and compared it to the allowed range of k_2 values that could be obtained from eccentricity measurements of HAT-P-13b. We use the example of HAT-P-13b to show the general relationships between the quantities temperature, envelope metallicity, core mass, and Love number of a planet. For any given k_2 value a maximum possible core mass can be determined. For HAT-P-13b we find Mcore < 27 ME, based on the latest eccentricity measurement. We are able to constrain both the envelope and bulk metallicity of HAT-P-13b to 1 -- 11 times stellar metallicity and the extension of the isothermal layer in the planet's atmosphere to 3 -- 44 bar. Assuming equilibrium tidal theory, we find lower limits on the tidal Q consistent with 10^3 - 10^5.

Lunar Gravity and Rotation Measurements by Japanese Lunar Landing Missions

Measurements of lunar gravity and rotation are important because they provide information of the physical state of the lunar interior. Previously only passive LLR (Lunar Laser Ranging) using CCR (corner cube reflectors) has been applied for the detailed study of lunar librations, rotation variability. As for candidate instruments for SELENE-2 (forthcoming lunar landing mission by JAXA) and future lunar missions, we propose VLBI (inverse-VLBI and differential VLBI) for gravity measurement to constrain tidal Love number, LLR (Lunar Laser Ranging) and ILOM (In-situ Lunar Orientation Measurement) for libration measurements.

Detecting tides and gravity at Europa from multiple close flybys

[1] This paper presents the expected accuracy of the tides and gravity of Europa that can be measured by tracking a spacecraft during close flybys of Europa. A reference trajectory was designed for flyby science observations and consists of a total of 36 flybys of Europa at 100 km altitude. Earth-based Doppler measurements were created during ±2 hours of each periapsis passage and were simulated with realistic dynamical and measurement assumptions. The result shows that the degree 2 tidal Love number,k2, can be estimated to σk2 = 0.045 and σk2= 0.009 (1-sigma formal uncertainty) assuming X-band and Ka-band tracking capabilities, respectively, which is sufficient to confirm the existence of a global subsurface ocean.

LONG-TERM STABLE EQUILIBRIA FOR SYNCHRONOUS BINARY ASTEROIDS

Synchronous binary asteroids may exist in a long-term stable equilibrium, where the opposing torques from mutual body tides and the binary YORP (BYORP) effect cancel. Interior of this equilibrium, mutual body tides are stronger than the BYORP effect and the mutual orbit semi-major axis expands to the equilibrium; outside of the equilibrium, the BYORP effect dominates the evolution and the system semi-major axis will contract to the equilibrium. If the observed population of small (0.1 - 10 km diameter) synchronous binaries are in static configurations that are no longer evolving, then this would be confirmed by a null result in the observational tests for the BYORP effect. The confirmed existence of this equilibrium combined with a shape model of the secondary of the system enables the direct study of asteroid geophysics through the tidal theory. The observed synchronous asteroid population cannot exist in this equilibrium if described by the canonical "monolithic" geophysical model. The "rubble pile" geophysical model proposed by \citet{Goldreich2009} is sufficient, however it predicts a tidal Love number directly proportional to the radius of the asteroid, while the best fit to the data predicts a tidal Love number inversely proportional to the radius. This deviation from the canonical and \citet{Goldreich2009} models motivates future study of asteroid geophysics. Ongoing BYORP detection campaigns will determine whether these systems are in an equilibrium, and future determination of secondary shapes will allow direct determination of asteroid geophysical parameters.

THERMAL EVOLUTION AND STRUCTURE MODELS OF THE TRANSITING SUPER-EARTH GJ 1214b

The planet GJ 1214b is the second known super-Earth with a measured mass and radius. Orbiting a quiet M-star, it receives considerably less mass-loss driving X-ray and UV radiation than CoRoT-7b, so that the interior may be quite dissimilar in composition, including the possibility of a large fraction of water. We model the interior of GJ 1214b assuming a two-layer (envelope+rock core) structure where the envelope material is either H/He, pure water, or a mixture of H/He and H2O. Within this framework we perform models of the thermal evolution and contraction of the planet. We discuss possible compositions that are consistent with Mp=6.55 ME, Rp=2.678 RE, an age tau=3-10 Gyr, and the irradiation level of the atmosphere. These conditions require that if water exists in the interior, it must remain in a fluid state, with important consequences for magnetic field generation. These conditions also require the atmosphere to have a deep isothermal region extending down to 80-800 bar, depending on composition. Our results bolster the suggestion of a metal-enriched H/He atmosphere for the planet, as we find water-world models that lack an H/He atmosphere to require an implausibly large water-to-rock ratio of more than 6:1. We instead favor a H/He/H2O envelope with high water mass fraction (~0.5-0.85), similar to recent models of the deep envelope of Uranus and Neptune. Even with these high water mass fractions in the H/He envelope, generally the bulk composition of the planet can have subsolar water:rock ratios. Dry, water-enriched, and pure water envelope models differ to an observationally significant level in their tidal Love numbers k2 of respectively ~0.018, 0.15, and 0.7.

Geodesy constraints on the interior structure and composition of Mars

Knowledge of the interior structure of Mars is of fundamental importance to the understanding of its past and present state as well as its future evolution. The most prominent interior structure properties are the state of the core, solid or liquid, its radius, and its composition in terms of light elements, the thickness of the mantle, its composition, the presence of a lower mantle, and the density of the crust. In the absence of seismic sounding only geodesy data allow reliably constraining the deep interior of Mars. Those data are the mass, moment of inertia, and tides. They are related to Mars’ composition, to its internal mass distribution, and to its deformational response to principally the tidal forcing of the Sun. Here we use the most recent estimates of the moment of inertia and tidal Love number k 2 in order to infer knowledge about the interior structure of the Mars. We have built precise models of the interior structure of Mars that are parameterized by the crust density and thickness, the volume fractions of upper mantle mineral phases, the bulk mantle iron concentration, and the size and the sulfur concentration of the core. From the bulk mantle iron concentration and from the volume fractions of the upper mantle mineral phases, the depth dependent mineralogy is deduced by using experimentally determined phase diagrams. The thermoelastic properties at each depth inside the mantle are calculated by using equations of state. Since it is difficult to determine the temperature inside the mantle of Mars we here use two end-member temperature profiles that have been deduced from studies dedicated to the thermal evolution of Mars. We calculate the pressure and temperature dependent thermoelastic properties of the core constituents by using equations state and recent data about reference thermoelastic properties of liquid iron, liquid iron–sulfur, and solid iron. To determine the size of a possible inner core we use recent data on the melting temperature of iron–sulfur. Within our model assumptions the geodesy data imply that Mars has no solid inner core and that the liquid core contains a large fraction of sulfur. The absence of a solid inner is in agreement with the absence of a global magnetic field. We estimate the radius of the core to be 1794 ± 65 km and its core sulfur concentration to be 16 ± 2 wt%. We also show that it is possible for Mars to have a thin layer of perovskite at the bottom of the mantle if it has a hot mantle temperature. Moreover a chondritic Fe/Si ratio is shown to be consistent with the geodesy data, although significantly different value are also possible. Our results demonstrate that geodesy data alone, even if a mantle temperature is assumed, can almost not constrain the mineralogy of the mantle and the crust. In order to obtain stronger constraints on the mantle mineralogy bulk properties, like a fixed Fe/Si ratio, have to be assumed.

Predictions on the core mass of Jupiter and of giant planets in general

More than 80 giant planets are known by mass and radius. Their interior structure in terms of core mass, number of layers, and composition however is still poorly known. An overview is presented about the core mass M core and envelope mass of metals M Z in Jupiter as predicted by various equations of state. It is argued that the uncertainty about the true H/He EOS in a pressure regime where the gravitational moments J 2 and J 4 are most sensitive, i.e. between 0.5 and 4 Mbar, is in part responsible for the broad range \(M_{\mathit{core}}=0{-}18\:M_{\oplus }\), \(M_{Z}=0{-}38\:M_{\oplus }\), and \(M_{\mathit{core}}+M_{Z}=14{-}38\:M_{\oplus }\) currently offered for Jupiter. We then compare the Jupiter models obtained when we only match J 2 with the range of solutions for the exoplanet \(\mathrm{GJ}\:436\mathrm{b}\), when we match an assumed tidal Love number k 2 value.