Tidal Theory Research Articles

Can one hear supercontinents in the tides of ocean planets?

Context.Recent observations and theoretical progress made about the history of the Earth-Moon system suggest that tidal dissipation in oceans primarily drives the long-term evolution of orbital systems hosting ocean planets. Particularly, they emphasise the key role played by the geometry of land-ocean distributions in this mechanism. However, the complex way continents affect oceanic tides still remains to be elucidated.Aims.In the present study, we investigate the impact of a single supercontinent on the tidal response of an ocean planet and the induced tidally dissipated energy.Methods.The adopted approach is based on the linear tidal theory. By simplifying the continent to a spherical cap of a given angular radius and position on the globe, we carried out a harmonic analysis of the whole planet’s tidal response including the coupling with the solid part due to ocean loading and self-attraction variations. In this framework, tidal flows are formulated analytically in terms of explicitly defined oceanic eigenmodes, as well as the resulting tidal Love numbers, dissipated power, and torque.Results.The analysis highlights the symmetry breaking effect of the continent, which makes the dependence of tidal quantities on the tidal frequency become highly irregular. The metric introduced to quantify this continentality effect reveals abrupt transitions between polar and non-polar configurations, and between small-sized and medium-sized continents. Additionally, it predicts that a continent similar to South America or smaller (~30° angular radius) does not qualitatively alter the tidal response of a global ocean regardless of its position on the planet.

On the orbital decay of the gas giant Kepler-1658b

ABSTRACT The gas giant Kepler-1658b has been inferred to be spiralling into its sub-giant F-type host star Kepler-1658a (KOI-4). The measured rate of change of its orbital period is $\stackrel{\bf \centerdot }{\textstyle {P}}_{\rm orb}\, =\, -\, 131^{+20}_{-22}\,\rm {ms\,yr^{ -1}}$, which can be explained by tidal dissipation in the star if its modified tidal quality factor is as low as $Q^{\, \prime }\approx 2.50\times {10}^{4}$. We explore whether this could plausibly be consistent with theoretical predictions based on applying up-to-date tidal theory in stellar models (varying stellar mass, age, and metallicity) consistent with our newly derived observational constraints. In most of our models matching the combined constraints on the stellar effective temperature and radius, the dissipation in the star is far too weak, capable of providing $Q^{\, \prime }\gtrsim 10^9$, hence contributing negligibly to orbital evolution. Using only constraints on the stellar radius, efficient tidal dissipation sufficient to explain observations is possible due to inertial waves in the convective envelope during the sub-giant phase, providing $Q^{\, \prime }\sim 10^4$, but this period in the evolution is very short-lived (shorter than 102 yr in our models). We show that dissipation in the planet is capable of explaining the observed $\dot{P}_\mathrm{orb}$ only if the planet rotates non-synchronously. Tidally induced pericentre precession is a viable explanation if the periastron argument is near 3π/2 and the planet's quadrupolar Love number is above 0.26. Further observations constraining the stellar and planetary properties in this system have the exciting potential to test tidal theories in stars and planets.

Slowly rotating close binary stars in Cassini states

ABSTRACT Recent asteroseismic measurements have revealed a small population of stars in close binaries, containing primaries with extremely slow rotation rates. Such stars defy the standard expectation of tidal synchronization in such systems, but they can potentially be explained if they are trapped in a spin-orbit equilibrium known as Cassini state 2 (CS2). This state is maintained by orbital precession due to an outer tertiary star, and it typically results in a very sub-synchronous rotation rate and high degree of spin-orbit misalignment. We examine how CS2 is affected by magnetic braking and different types of tidal dissipation. Magnetic braking results in a slower equilibrium rotation rate, while tidal dissipation via gravity waves can result in a slightly higher rotation rate than predicted by equilibrium tidal theory, and dissipation via inertial waves can result in much slower rotation rates. For seven binary systems with slowly rotating primaries, we predict the location of the outer tertiary predicted by the CS2 theory. In five of these systems, a tertiary companion has already been detected, although it is closer than expected in three of these, potentially indicating tidal dissipation via inertial waves. We also identify a few new candidate systems among a population of eclipsing binaries with rotation measurements via spot modulation.

The role of tidal interactions in the formation of slowly rotating early-type stars in young star clusters

ABSTRACT The split main sequences found in the colour–magnitude diagrams of star clusters younger than ∼600 Myr are suggested to be caused by the dichotomy of stellar rotation rates of upper main-sequence stars. Tidal interactions have been suggested as a possible explanation of the dichotomy of the stellar rotation rates. This hypothesis proposes that the slow rotation rates of stars along the split main sequences are caused by tidal interactions in binaries. To test this scenario, we measured the variations in the radial velocities of slowly rotating stars along the split main sequence of the young Galactic cluster NGC 2422 (∼90 Myr) using spectra obtained at multiple epochs with the Canada–France–Hawai’i Telescope. Our results show that most slowly rotating stars are not radial velocity variables. Using the theory of dynamical tides, we find that the binary separations necessary to fully or partially synchronize our spectroscopic targets, on time-scales shorter than the cluster age, predict much larger radial velocity variations across multiple-epoch observations, or a much larger radial velocity dispersion at a single epoch, than the observed values. This indicates that tidal interactions are not the dominant mechanism to form slowly rotating stars along the split main sequences. As the observations of the rotation velocity distribution among B- and A-type stars in binaries of larger separations hint at a much stronger effect of braking with age, we discuss the consequences of relaxing the constraints of the dynamical tides theory.

Thermal Tides on Mars Before and During the 2018 Global Dust Event as Observed by TIRVIM‐ACS Onboard ExoMars Trace Gas Orbiter

AbstractMigrating tides dominate the tropical climate on Mars and are known to reach high amplitudes during global dust events (GDE). In this study, we characterize the amplitude, phase and vertical wavelength of the diurnal and semidiurnal migrating tides in Mars' lower atmosphere (up to 50 km) by exploiting temperature vertical profiles retrieved from TIRVIM, an infrared spectrometer onboard the ExoMars Trace Gas Orbiter covering multiple local times. Observations from the Mars Climate Sounder onboard the Mars Reconnaissance Orbiter are used to complement the local time coverage when needed, and to estimate a seasonal trend to subtract from TIRVIM observations. We focus on two time periods in Martian Year 34, near Ls = 150° and near Ls = 200° (during the 2018 GDE). The characteristics of the migrating tides at Ls = 150° agree very well with tidal theory: a downward propagation, amplitudes of typically 2–5 K, and a larger vertical wavelength for the semidiurnal compared to the diurnal mode. Comparisons with model predictions from the Mars Planetary Climate Model reveal an excellent agreement, except for a slightly different phase of the diurnal tide. During the GDE, the tide pattern changes spectacularly: the diurnal tide amplitude reaches 35 K at 65°S and 17 K at 50°N, being vertically trapped up to 10 Pa. The semidiurnal tide is maximum near 20–30°S with an amplitude of 8–12 K. The phase of this mode is tilted with latitude, which was not the case before the storm. This indicates a significant contribution of the asymmetric Hough modes due to hemispheric asymmetry in the dust distribution.

Detailed equilibrium and dynamical tides: impact on circularization and synchronization in open clusters

ABSTRACT Binary stars evolve into chemically peculiar objects and are a major driver of the galactic enrichment of heavy elements. During their evolution they undergo interactions, including tides, that circularize orbits and synchronize stellar spins, impacting both individual systems and stellar populations. Using Zahn’s tidal theory and mesa main-sequence model grids, we derive the governing parameters λlm and E2, and implement them in the new mint library of the stellar population code binary_c. Our mint equilibrium tides are two to five times more efficient than the ubiquitous bse prescriptions, while the radiative-tide efficiency drops sharply with increasing age. We also implement precise initial distributions based on bias-corrected observations. We assess the impact of tides and initial orbital-parameter distributions on circularization and synchronization in eight open clusters, comparing synthetic populations and observations through a bootstrapping method. We find that changing the tidal prescription yields no statistically significant improvement as both calculations typically lie within 0.5σ. The initial distribution, especially the primordial concentration of systems at log10(P/d) ≈ 0.8, e ≈ 0.05 dominates the statistics even when artificially increasing tidal strength. This confirms the inefficiency of tides on the main sequence and shows that constraining tidal-efficiency parameters using the e − log10(P/d) distribution alone is difficult or impossible. Orbital synchronization carries a more striking age-dependent signature of tidal interactions. In M35 we find twice as many synchronized rotators in our mint calculation as with bse. This measure of tidal efficiency is verifiable with combined measurements of orbital parameters and stellar spins.

Short-Term Variability of Non-Migrating Diurnal Tides in the Stratosphere from CMAM30, ERA-Interim, and FORMOSAT-3/COSMIC

The variability of non-migrating tides in the stratosphere is investigated using temperature data from Canadian Middle Atmosphere Model (CMAM30), ERA-interim reanalysis and Formosa Satellite-3 and Constellation Observing System for Meteorology, Ionosphere, and Climate (FORMOSAT-3/COSMIC) from 2006 to 2010 using a ±10-day window. CMAM30 and ERA results show that the amplitudes of non-migrating tides, DS0 and DW2, are negligible in the mid and high-latitude stratosphere, and the results from satellite datasets are significantly affected by aliasing in this region, in spite of using a smaller window size for analysis (±10 days). Significant short term variability ranging from 30 to 100 days is observed in DS0 and DW2 over the equatorial and tropical latitudes. These tides are seen as two prominent bands around the equator with DS0 maximising during boreal summers and DW2 maximising during boreal winters. These variabilities are compared with the variability in amplitude of the stationary planetary wave with wavenumber one (SPW1) in the high-latitude stratosphere using the continuous wavelet transform (CWT). It is found that during boreal winters, the variability of SPW1 at 10 hPa over 65° N is similar to that of DS0 and DW2 over the equator at 0.0007 hPa. This provides evidence that SPW1 from the high-altitude stratosphere moving upward and equator-ward could be interacting with the migrating diurnal tide and generating the non-migrating tides in the equatorial mesosphere and lower thermosphere (MLT). The variabilities, however, are not comparable during summers, with SPW1 being absent in the Northern Hemisphere. It is thus concluded that non-linear interactions could be a source of non-migrating tidal variability in the equatorial MLT region during boreal winters, but during summers, the tidal variabilities have other sources in the lower atmosphere. The anti-symmetric nature of the vertical global structures indicates that these tides could be the result of global atmospheric oscillations proposed by the classical tidal theory.

A Tidal Theory Based on the Inertial Motion of the Matter in the Universe

A Tidal Theory Based on the Inertial Motion of the Matter in the Universe

The 27.3-day oscillation in the lunar tidal theory and in atmospheric data

In the history of atmospheric science, the lunar tropical-monthly oscillation with the period of 27.32 days in atmospheric data has been discovered several times. However, up to now, analysis of possible influence of other lunar tidal oscillations on the extracted tropical-monthly oscillation was not provided. The historical and modern discoveries of this tidal oscillation in the atmosphere are reconsidered in the present paper. Although its existence in the atmosphere was predicted by the lunar tidal theory, its amplitude must be roughly one order less than that of the lunar anomalistic tidal oscillation having a very close period of 27.55 days. Moreover, in spite of the different periods these two oscillations appear to be mutually dependent. The analysis of the previous findings demonstrates that, in some papers, the tropical monthly oscillation was not properly separated from the anomalistic monthly oscillation and hence the presence of the tropical monthly oscillation in atmospheric data was not convincing. Krahenbuhl et al. (2011) were first who managed to extract convincingly the measurable tropical monthly oscillation in atmospheric data. Now the existence of the lunar tropical monthly oscillation in the atmosphere should be regarded as a solved problem.

Progress report on the BEBOP search for circumbinary planets with radial velocities

AbstractThe BEBOP (Binaries Escorted By Orbiting Planets) survey is a search for circumbinary planets using the radial velocity spectrographs HARPS and SOPHIE, currently focusing on single-lined binaries with a mass ratio < 0.3. Circumbinary systems are an important testing ground for planet formation theories as the dynamically complex influence of the binary makes planet formation and survival more difficult. Here we present the results of the survey so far including: confirmed planets such as BEBOP-1c the first circumbinary planet detected in radial velocity; the status of our observations; and preliminary occurrence rates. We compare the early results of the radial velocity survey to the population of circumbinary planets discovered in transit, and suggest that there may be a population of inflated planets close to the inner binary which are detectable in transit but more difficult in radial velocity. Using time-lag tidal theory, we show that this inflation is unlikely caused by tides.

The Distant Action of the Heavens in Girolamo Borri’s Tidal Theory

Abstract The Aristotelian professor of natural philosophy and courtier at the Medici in Florence, Girolamo Borri, developed a theory based on heat to explain the tidal motions of the sea. In his dialogues on this phenomenon, he deemed that tides follow from the ‘moderate’ simmering of the waters as an effect of lunar light. His tidal theory displaced the theory of the Moon’s distant action on terrestrial waters from its traditionally astrological connotation. Moreover, his theory was not ‘empirical’ but rather inserted in a broad natural philosophical and cosmological framework. Although Galileo Galilei later dismissed heat-based explanations of the tides, such explanations are historically relevant as part of the larger scientific picture, in which controversies over the phenomenon of the tides of the sea fuelled cosmological, even post-Copernican assessments of the connection between terrestrial physics and incipient celestial physics.

Jupiter's Banding and Jets May Be Caused by Tides

The stratification parameters for Jupiter’s outer 3000 km shell are calculated using a density profile recently derived from observations of the NASA Juno spacecraft currently in Jovian orbit. Using these parameters, the equations of classical tidal theory for a stratified, nonhydrostatic, compressible fluid are numerically solved for sectoral tidal forcing by Io. The results support a long-standing though little discussed proposal that the banding/jets (and possibly the unexplained endogenic heat) are caused by the tides. First, general arguments from eigenmode analyses expect resonantly forced tidal modes and the scattering of the tidal response to higher spatial degrees by Jupiter’s fast rotation, with time-averaged tidal effects appearing in bands between critical latitudes (±50° for forcing by Io). Second, resonant tides and banding are specifically demonstrated in the tidal model configured with the Juno-derived stratification. While banding in the time-averaged tidal features is a robust expectation (from the well-prescribed forcing and rotation parameters) and is independent of the internal parameters, the details of the banding (e.g., number, width) are highly dependent. Hence, comparison of the tidal model with observations provides a test of the tidal hypothesis of the bands as well as assumptions of interior parameters and processes. Here, dissipation parameterized as a simple pressure relaxation term in the vertical balance equation shows a time-mean banded structure between the critical latitudes that can drive geostrophic jets matching the major observed features, including strong prograde flow at the equator. By contrast, alternate stratification/dissipation assumptions produce banded structures that do not agree with observations.

Orbital evolution of circumbinary planets due to creep tides

Most confirmed circumbinary planets are located very close to their host binary where the tidal forces are expected to play an important role in their dynamics. Here we consider the orbital evolution of a circumbinary planet with arbitrary viscosity, subjected to tides due to both central stars. We adopt the creep tide theory and assume that the planet is the only extended body in the system and that its orbital evolution occurs after acquiring its pseudo-synchronous stationary rotational state. With this aim, we first performed a set of numerical integrations of the tidal equations, using a Kepler-38-type system as a working example. For this case we find that the amount of planetary tidal migration and also, curiously, its direction both depend on the viscosity. However, the effect of tides on its eccentricity and pericenter evolutions is simply a move toward pure gravitational secular solutions. Then we present a secular analytical model for the planetary semimajor axis and eccentricity evolution that reproduces very well the mean behavior of the full tidal equations and provides a simple criterion to determine the migration directions of the circumbinary planets. This criterion predicts that some of the confirmed circumbinary planets are tidally migrating inward, but others are migrating outward. However, the typical timescales are predicted to be very long, and not much orbital tidal evolution is expected to have taken place in these systems. Finally, we revisit the orbital evolution of a circumbinary planet in the framework of the constant time lag model. We find that the results predicted with this formalism are identical to those obtained with creep theory in the limit of gaseous bodies.

Tidally induced migration of TESS gas giants orbiting M dwarfs

ABSTRACT According to core-accretion formation models, the conditions under which gas giants will form around M dwarfs are very restrictive. Also, the correlation of the occurrence of these planets with the metallicity of host stars is still unknown due to the intrinsic faintness of M dwarfs in the optical and some intricacies in their spectra. Interestingly, NASA’s Transiting Exoplanet Survey Satellite (TESS) mission has started to create a growing sample of these systems, with 11 observed planets located in close-in orbits: contrary to what is expected for low-mass stars. Tidal interactions with the host star will play a key role in determining the fate of these planets, so by using the measured physical and orbital characteristics of these M-dwarf systems we numerically analyse the exchange of rotational and orbital angular momentum, while constraining the energy dissipation in each system to calculate whether host stars are spun up or spun down, depending on the relationship between the gain and loss of angular momentum by the stellar rotation. We also study the coupled orbital and physical evolution of their gas giant companion and calculate orbital circularization time-scales, as well as the time needed to undergo orbital decay from their current orbital position to the Roche limit. The thorough study of tidal processes occurring over short and long time-scales in star–planet systems like those studied here, can help constrain tidal dissipation rates inside the star and planet, complement tidal theories, and improve estimations of unconstrained properties of exoplanetary systems.

Magnetic effect on equilibrium tides and its influence on the orbital evolution of binary systems

In the standard theory of equilibrium tides, hydrodynamic turbulence is considered. In this paper we study the effect of magnetic fields on equilibrium tides. We find that the turbulent Ohmic dissipation associated with a tidal flow is much stronger than the turbulent viscous dissipation such that a magnetic field can greatly speed up the tidal evolution of a binary system. We then apply the theory to three binary systems: the orbital migration of 51 Pegasi b, the orbital decay of WASP-12b, and the circularization of close binary stars. Theoretical predictions are in good agreement with observations, which cannot be clearly interpreted with hydrodynamic equilibrium tides.

Anomalous 18.61-Year Nodal Cycles in the Gulf of Tonkin Revealed by Tide Gauges and Satellite Altimeter Records

Understanding nodal tidal characteristics is essential for accurate long-term tidal prediction. Observational nodal evolution of tides is mainly based on tide gauge records in coastal areas which are limited in time and space, thus impeding coherent determinations of basin-wide patterns of tidal variability. In this paper, we indicate the potential of satellite altimeter data to investigate 18.61-year nodal modulations of main constituents in the Gulf of Tonkin. Three tide gauges and multi-source satellite altimeter observations (TOPEX/Poseidon, Jason1, Jason2, and Jason3) revealed that 18.61-year nodal cycles in tidal amplitudes have noticeable deviations from the equilibrium tidal theory in the Gulf of Tonkin. In general, M2 and N2 nodal modulations are anomalously larger than theoretical values while K2, K1, and O1 nodal modulations are noticeably smaller than theoretical values. Compared to point-based tide gauges, satellite altimeter records can provide basin-wide features of nodal modulations of main constituents. Although overlapping geographical blocks are applied to eliminate the effect of tidal alias originated from long-period sampling intervals, the estimation of nodal cycles of minor constituents are still questionable. Nevertheless, the methods described here provide a strong foundation for future research on time-varying tidal dynamics using the combination of tide gauges and satellite altimeter data.

Atmospheric Gravitational Tides of Earth-like Planets Orbiting Low-mass Stars

Temperate terrestrial planets orbiting low-mass stars are subject to strong tidal forces. The effects of gravitational tides on the solid planet and that of atmospheric thermal tides have been studied, but the direct impact of gravitational tides on the atmosphere itself has so far been ignored. We first develop a simplified analytic theory of tides acting on the atmosphere of a planet. We then implement gravitational tides into a general circulation model of a static-ocean planet in a short-period orbit around a low-mass star—the results agree with our analytic theory. Because atmospheric tides and solid-body tides share a scaling with the semimajor axis, we show that there is a maximum amplitude of the atmospheric tide that a terrestrial planet can experience while still having a solid surface; Proxima Centauri b is the poster child for a planet that could be geophysically Earth-like but with atmospheric tides more than 500× stronger than Earth’s. In this most extreme scenario, we show that atmospheric tides significantly impact the planet’s meteorology—but not its climate. Two possible modest climate impacts are enhanced longitudinal heat transport and cooling of the lowest atmospheric layers. The strong radiative forcing of such planets dominates over gravitational tides, unlike moons of cold giant planets, such as Titan. We speculate that atmospheric tides could be climatologically important on planets where the altitude of maximal tidal forcing coincides with the altitude of cloud formation and that the effect could be detectable for non-Earth-like planets subject to even greater tides.

Ocean Tides near Hawaii from Satellite Altimeter Data. Part II

Abstract In Part I, the Chebyshev polynomial fitting (CPF) method has been proved to be effective to construct reliable cotidal charts for the eight major tidal constituents (M2, S2, K1, O1, N2, K2, P1, and Q1) near Hawaii and yields accurate results which are consistent with the Finite Element Solutions 2014 (FES2014), National Astronomical Observatory 99b (NAO.99b), and TPXO9 models. In this paper, the method is extended to estimate the harmonic constants of some minor tidal constituents. The mesoscale variation correction is applied to tidal elevations from satellite altimeters to eliminate the potential influence of background mesoscale ocean noise when estimating minor tidal constituents. This correction is necessary and makes the amplitude ratio between P1 and K1 constituents more consistent with the equilibrium tidal theory. Compared with the harmonic constants directly extracted from satellite altimeter data, FES2014 and NAO.99b yield mean root-mean-square (RMS) errors of 0.238 and 0.226 cm, respectively, while CPF method yields a mean RMS error of 0.210 cm, causing a 7%–12% decrease in the RMS error. At the crossover points between ascending and descending tracks, the decrease of RMS errors becomes 15%–18%. The accuracy of this method is also validated by comparing the estimated harmonic constants with those derived from tidal gauges and bottom-pressure recorders. These results indicate that the CPF method is also effective for estimating harmonic constants of minor tidal constituents. More importantly, the CPF method can obtain the harmonic constants of minor tidal constituents directly from satellite altimeter data, instead of being inferred via admittance theory. Significance Statement Ocean tides originate from the gravitational attraction of the sun and moon. Among the large number of tidal constituents, the major tidal constituents have been extensively studied. We extend the method proposed in Part I to estimate the harmonic constants of some minor tidal constituents with low amplitudes. The method relies on actual observations of water level variations from satellite altimeters without considering the hydrodynamic equations. We compared the results of this method with those of other models, using the data from satellite altimeters, tidal gauges, and bottom-pressure recorders. We find that the method performs well in estimating harmonic constants for some minor tidal constituents and causes a decrease of 7%–18% in RMS errors compared to other models.

Do tides destabilize Trojan exoplanets?

One outstanding problem in extrasolar planet studies is why no co-orbital exoplanets have been found, despite numerous searches among the many known planetary systems, many of them in other mean-motion resonances. Here we examine the hypothesis that dissipation of energy by tides in Trojan planets is preventing their survival.The Appendix of this paper generalizes the conventional theory of tides to include tidal forces independent of dissipation, as well as the effects of one body on tides raised by another. The main text applies this theory to a model system consisting of a primary of stellar mass, a secondary of sub-stellar mass in a circular orbit about the primary, and a much lighter Trojan planet librating with small amplitude about an equilateral point of the system.Next, we linearize the equations of motion about the Trojan points, including the tidal forces, and solve for the motion of the Trojan. The results indicate that tides damp out the Trojan’s motion perpendicular to the orbital plane of the primary and secondary, as well as its epicycles due to its eccentricity; but they pump up the amplitude of its tadpole librations exponentially. We then verify our analytic solutions by integrating the non-linearized equations of motion numerically for several sample cases. In each case, we find that the librations grow until the Trojan escapes its libration, which leads to a close encounter with either the primary or the secondary.

Creep tide theory: equations for differentiated bodies with aligned layers

The creep tide theory is used to establish the basic equations of the tidal evolution of differentiated bodies formed by aligned homogeneous layers in co-rotation. The mass concentration of the body is given by the fluid Love number $$k_f$$ . The formulas are given by series expansions valid for high eccentricity systems. They are equivalent to Darwin’s equations, but formally more compact. An application to the case of Enceladus, with $$k_f=0.942$$ , is discussed.