Tidal Theory Research Articles

The GAPS Programme with HARPS-N at TNG

We carried out a Bayesian homogeneous determination of the orbital parameters of 231 transiting giant planets (TGPs) that are alone or have distant companions; we employed DE-MCMC methods to analyse radial-velocity (RV) data from the literature and 782 new high-accuracy RVs obtained with the HARPS-N spectrograph for 45 systems over 3 years. Our work yields the largest sample of systems with a transiting giant exoplanet and coherently determined orbital, planetary, and stellar parameters. We found that the orbital parameters of TGPs in non-compact planetary systems are clearly shaped by tides raised by their host stars. Indeed, the most eccentric planets have relatively large orbital separations and/or high mass ratios, as expected from the equilibrium tide theory. This feature would be the outcome of high-eccentricity migration (HEM). The distribution of $\alpha=a/a_R$, where $a$ and $a_R$ are the semi-major axis and the Roche limit, for well-determined circular orbits peaks at 2.5; this also agrees with expectations from the HEM. The few planets of our sample with circular orbits and $\alpha >5$ values may have migrated through disc-planet interactions instead of HEM. By comparing circularisation times with stellar ages, we found that hot Jupiters with $a < 0.05$ au have modified tidal quality factors $10^{5} < Q'_p < 10^{9}$, and that stellar $Q'_s > 10^{6}-10^{7}$ are required to explain the presence of eccentric planets at the same orbital distance. As a by-product of our analysis, we detected a non-zero eccentricity for HAT-P-29; we determined that five planets that were previously regarded to have hints of non-zero eccentricity have circular orbits or undetermined eccentricities; we unveiled curvatures caused by distant companions in the RV time series of HAT-P-2, HAT-P-22, and HAT-P-29; and we revised the planetary parameters of CoRoT-1b.

The pole tide/14-month oscillations in the Baltic Sea during the 19th and 20th centuries: Spatial and temporal variations

Sea level changes with a period of 14 months (P14) are usually associated with the pole tide, which is the oceanic response to the Chandler wobble in the Earth's axis of rotation. The amplitudes of these changes in the Baltic Sea are anomalously large, much larger than follows from the equilibrium pole tide theory. It appears that the oscillations are related to meteorological forcing rather than to pole motions. To examine the P14 properties, we used long-term (48–213 years) tide gauge records from 77 stations located in the Baltic Sea and adjacent area of the North Sea. High-resolution sea level spectra revealed a cluster of peaks with periods from 410 to 450 days. The temporal variations in the P14 sea level amplitudes and peak frequencies are found to be considerable and poorly correlated with the modulation of the Earth's pole motions. In contrast, our findings for 1871–2011 demonstrate strong resemblance between temporal (year-to-year) variations of the P14 peak and zonal wind, confirming earlier results of Ekman (1996) and O’Connor et al. (2000), who indicated the major role of the zonal wind in formation of the 14-month oscillations in the Baltic and North seas. The integral amplitudes of the P14 oscillations in the Baltic Sea gradually increase eastward from the entrance (the Danish straits) to the farthest ends of the sea with the largest amplitudes (up to 6.5cm) observed at the very head of the Gulf of Finland. Similar P14 amplification toward the west coast of Denmark is also observed along the south coast of the North Sea. The geographical distribution of the P14 in these two regions are almost the same as of the seasonal sea level oscillations, indicating presumably similar formation mechanisms. Extensive shallow-water areas in the Baltic Sea and southeastern part of the North Sea, combined with intense meteorological forcing, appear to be two key factors responsible for the formation of anomalously strong 14-month oscillations in these regions.

Odd Harmonics in Exoplanet Photometry: Weather or Artifact?

Photometry of short-period planetary systems allows astronomers to monitor exoplanets, their host stars, and their mutual interactions. In addition to the transits of a planet in front of its star and the eclipses of the planet by its star, researchers have reported flux variations at the orbital frequency and its harmonics: planetary reflection and/or emission and Doppler beaming of starlight produce one peak per orbit, while ellipsoidal variations of a tidally distorted star and/or planet produce two maxima per orbit. Researchers have also reported significant photometric variability at three times the orbital frequency, sometimes much greater than the predictions of tidal theory. The reflected phase variations of a homogeneous planet contains power at even orbital harmonics-important for studies of ellipsoidal variations-but cannot contain odd orbital harmonics. We show that odd harmonics can, however, be produced by an edge-on planet with a time-variable map, or an inclined planet with a North-South (N-S) asymmetric map. Either of these scenarios entail weather: short-period planets are expected to have zero obliquity and hence N-S symmetric stellar forcing. North-South asymmetry in a giant planet would therefore suggest stochastic localized features, such as weather. However, we find that previous claims of large-amplitude odd modes in Kepler photometry are artifacts of removing planetary transits rather than modeling them. The only reliable claims of odd harmonics remain HAT-P-7b and Kepler-13Ab, for which the third mode amplitude is 6-8% of the planetary flux. Although time-variable albedo maps could in principle explain these odd harmonics, upper-limits on the infrared variability of hot Jupiters make this hypothesis unlikely. We urge theorists to study the effects of close-in planets on stellar atmospheres, as this remains the only plausible hypothesis.

TIDALLY INDUCED PULSATIONS IN KEPLER ECLIPSING BINARY KIC 3230227

ABSTRACT KIC 3230227 is a short period (P ≈ 7.0 days) eclipsing binary with a very eccentric orbit (e = 0.6). From combined analysis of radial velocities and Kepler light curves, this system is found to be composed of two A-type stars, with masses of M 1 = 1.84 ± 0.18 M ⊙, M 2 = 1.73 ± 0.17 M ⊙ and radii of R 1 = 2.01 ± 0.09 R ⊙, R 2 = 1.68 ± 0.08 R ⊙ for the primary and secondary, respectively. In addition to an eclipse, the binary light curve shows a brightening and dimming near periastron, making this a somewhat rare eclipsing heartbeat star system. After removing the binary light curve model, more than 10 pulsational frequencies are present in the Fourier spectrum of the residuals, and most of them are integer multiples of the orbital frequency. These pulsations are tidally driven, and both the amplitudes and phases are in agreement with predictions from linear tidal theory for l = 2, m = −2 prograde modes.

TIDAL RESPONSE OF PRELIMINARY JUPITER MODEL

ABSTRACT In anticipation of improved observational data for Jupiter’s gravitational field, from the Juno spacecraft, we predict the static tidal response for a variety of Jupiter interior models based on ab initio computer simulations of hydrogen–helium mixtures. We calculate hydrostatic-equilibrium gravity terms, using the non-perturbative concentric Maclaurin Spheroid method that eliminates lengthy expansions used in the theory of figures. Our method captures terms arising from the coupled tidal and rotational perturbations, which we find to be important for a rapidly rotating planet like Jupiter. Our predicted static tidal Love number, , is ∼10% larger than previous estimates. The value is, as expected, highly correlated with the zonal harmonic coefficient J 2, and is thus nearly constant when plausible changes are made to the interior structure while holding J 2 fixed at the observed value. We note that the predicted static k 2 might change, due to Jupiter’s dynamical response to the Galilean moons, and find reasons to argue that the change may be detectable—although we do not present here a theory of dynamical tides for highly oblate Jovian planets. An accurate model of Jupiter’s tidal response will be essential for interpreting Juno observations and identifying tidal signals from effects of other interior dynamics of Jupiter’s gravitational field.

Coupled orbital and spin evolution of the CoRoT-7 two-planet system using a Maxwell viscoelastic rheology

We investigate the orbital and rotational evolution of the CoRoT-7 two-planet system, assuming that the innermost planet behaves like a Maxwell body. We numerically resolve the coupled differential equations governing the instantaneous deformation of the inner planet together with the orbital motion of the system. We show that, depending on the relaxation time for the deformation of the planet, the orbital evolution has two distinct behaviours: for relaxation times shorter than the orbital period, we reproduce the results from classic tidal theories, for which the eccentricity is always damped. However, for longer relaxation times, the eccentricity of the inner orbit is secularly excited and can grow to high values. This mechanism provides an explanation for the present high eccentricity observed for CoRoT-7 b, as well as for other close-in super-Earths in multiple planetary systems.

An Analytical Approach to Determining Resonance in Semi-Closed Convergent Tidal Channels

An analytical model is used to investigate the resonant behavior in a semi-closed channel. The main integral quantities of the tidal wave are obtained by means of a linearized one-dimensional model as a function of three dimensionless parameters, representing cross-section convergence, friction and distance to the closed boundary. Arbitrary along-channel variations of width and depth are accounted for by using a multi-reach approach, whereby the main tidal dynamics are reconstructed by solving a set of linear equations satisfying the continuity conditions of water level and discharge at the junctions of the sub-reaches. We highlight the importance of depth variation in the momentum equation, which is not considered in the classical tidal theory. The model allows for a direct characterization of the resonant response and for the understanding of the relative importance of the controlling parameters, highlighting the role of convergence and friction. Subsequently, the analytical model is applied to the Bristol Channel and the Guadalquivir estuary. The proposed analytical relations provide direct insights into the tidal resonance in terms of tidal forcing, geometry and friction, which will be useful for the study of semi-closed tidal channels that experience relatively large tidal ranges at the closed end.

Tidal-Fluvial and Estuarine Processes in the Lower Columbia River: II. Water Level Models, Floodplain Wetland Inundation, and System Zones

Spatially varying water-level regimes are a factor controlling estuarine and tidal-fluvial wetland vegetation patterns. As described in Part I, water levels in the Lower Columbia River and estuary (LCRE) are influenced by tides, river flow, hydropower operations, and coastal processes. In Part II, regression models based on tidal theory are used to quantify the role of these processes in determining water levels in the mainstem river and floodplain wetlands, and to provide 21-year inundation hindcasts. Analyses are conducted at 19 LCRE mainstem channel stations and 23 tidally exposed floodplain wetland stations. Sum exceedance values (SEVs) are used to compare wetland hydrologic regimes at different locations on the river floodplain. A new predictive tool is introduced and validated, the potential SEV (pSEV), which can reduce the need for extensive new data collection in wetland restoration planning. Models of water levels and inundation frequency distinguish four zones encompassing eight reaches. The system zones are the wave- and current-dominated Entrance to river kilometer (rkm) 5; the Estuary (rkm-5 to 87), comprised of a lower reach with salinity, the energy minimum (where the turbidity maximum normally occurs), and an upper estuary reach without salinity; the Tidal River (rkm-87 to 229), with lower, middle, and upper reaches in which river flow becomes increasingly dominant over tides in determining water levels; and the steep and weakly tidal Cascade (rkm-229 to 234) immediately downstream from Bonneville Dam. The same zonation is seen in the water levels of floodplain stations, with considerable modification of tidal properties. The system zones and reaches defined here reflect geological features and their boundaries are congruent with five wetland vegetation zones.

Can brown dwarfs survive on close orbits around convective stars?

Brown dwarfs straddle the mass range transition from planetary to stellar objects. There is a relative paucity of brown dwarfs companions around FGKM stars compared to exoplanets for orbital periods less than a few years, but most of the short-period brown dwarf companions fully characterised by transits and radial velocities are found around F-type stars. We examine the hypothesis that brown dwarf companions could not survive on close orbit around stars with important convective envelopes because the tides and angular momentum loss through magnetic breaking should lead to a rapid orbital decay and quick engulfment of the companion. We use a classical Skumanich-type braking law, and constant time-lag tidal theory to assess the characteristic timescale for orbital decay for the brown dwarf mass range as a function of the host properties. We find that F-type stars may host massive companions for a significantly longer time than G-type stars for a given orbital period, which may explain the paucity of G-type hosts for brown dwarfs with orbital period less than 5 days. On the other hand, we show that the small radius of early M-type stars contributes to orbital decay timescales that are only half those of F-type stars, despite their more efficient tidal dissipation and magnetic braking. For fully convective later type M-dwarfs, orbital decay timescales could be orders of magnitude greater than for F-type stars. For orbital periods greater than 10 days, brown dwarf occurrence should largely be unaffected by tidal decay, whatever the mass of the host. On closer orbital periods, the rapid engulfment of massive companions could explain the lack of G and K-type hosts in the sample of known systems with transiting brown dwarfs. However, the paucity of M-type hosts can not be an effect of tidal decay alone, but may be the result of a selection effect in the sample and/or the formation mechanism.

Dynamic Elastic Tides

This is an exploration of dynamic tides on elastic bodies. The body is thought of as a dynamical system described by its modes of oscillation. The dynamics of these modes are governed by differential equations that depend on the rheology. The modes are damped by dissipation. Tidal friction occurs as exterior bodies excite the modes and the modes act back on the tide raising body. The whole process is governed by a closed set of differential equations. Standard results from tidal theory are recovered in a two-timescale approximation to the solution of these differential equations.

Latitude dependence of amplitude factor δ for degree 2 tides

The study updates the theory of tides by specifying latitude dependence of the gravimetric factor (amplitude factor δ) for diurnal and semidiurnal tidal waves in the oceanless elastic Earth. The strain of an elastic flattened gravitating sphere with latitude-dependent geopotential, density, and Lamé constants is described by a system of sixth-order ordinary differential equations. The elliptical sphere is presented as consisting of layers, with density and elasticity varying as a function of latitude along the sphere surface as mean radiuses of equidensity and equielasticity ellipsoids that cross this surface. Dissipation is taken into account as a logarithmic creep function. Integration of the derived equations allows avoiding the use of approximate methods. The predicted δ factors for degree 2 tides depend on latitude and increase from the equator to the pole by 0.12 to 0.18% for different Earth models. The obtained latitude dependences are compared with superconducting gravimeter (SG) data after including effects associated with the inertial and Coriolis forces, as well as dynamic resonance. The theoretical predictions and the observations show good fit, to hundredths of percent. The average δ factors predicted in this study for the PREM model are intermediate between those computed with the hydrostatic and nonhydrostatic tidal gravity models of Dehant, Defraigne, and Wahr (DDW/H and DDW/NH), while the estimates obtained with reference to the IASP91 model coincide with the DDW/NH results to the fifth decimal digit.

Tides in the mesopause region over Antarctica: Comparison of whole atmosphere model simulations with ground‐based observations

AbstractAlmost a quarter century ago first optical and radar observations from the South Pole revealed rich dynamics unexpected from classical tidal theory. A strong semidiurnal wind oscillation was detected near the mesopause implying substantial deviations from the classical view that the semidiurnal variation is dominated by the migrating tide. Subsequent systematic observations exhibited large seasonal variations of both the diurnal and semidiurnal tide with dramatic reduction in amplitude from summer to winter. First numerical simulations with a realistic general circulation model extending into the lower thermosphere indicated the presence of nonmigrating tides with substantial amplitudes in the polar regions. However, direct model‐data comparisons have been limited to idealized linear models. Here whole atmosphere model (WAM) simulations for January and July are compared with available wind climatologies based on multiyear radar observations at different locations in Antarctica as well as with first summertime lidar measurements of temperature. The diurnal tide simulation agrees well with most of the independent radar and satellite wind observations in both seasons. The strong semidiurnal tide comprised of migrating and nonmigrating components is well reproduced in summer, while in winter the model tends to overestimate the amplitudes over the continental edge. Besides model validation, a self‐consistent numerical solution also enables cross validation of observations made with different instruments at different locales.

Rainfall variations induced by the lunar gravitational atmospheric tide and their implications for the relationship between tropical rainfall and humidity

AbstractClassical tidal theory predicts that the lunar gravitational semidiurnal tide (L2) should induce perturbations in relative humidity (RH). Adiabatic expansion in divergent flow in advance of the L2 pressure minimum cools the air and reduces its saturation vapor pressure, thereby increasing the rate of condensation in saturated air parcels and causing the relative humidity (RH) of unsaturated parcels to rise. Here we detect a statistically significant L2 signature in precipitation rate (P) in a 15 year, eight times daily, global precipitation data set acquired in the Tropical Rainfall Measuring Mission. Consistent with tidal theory and with the notion that L2 modulates P mainly by perturbing RH, P varies out of phase with pressure, and it increases at a rate of about 10% per 1% increase in RH. These measurements thus provide a measure of the sensitivity of P to planetary‐scale changes in RH. Analysis of solar semidiurnal (S2) tidal statistics yields similar results.

Discussion of Hansen, J.M.; Aagaard, T., and Kuijpers, A., 2015. Sea-Level Forcing by Synchronization of 56- and 74-Year Oscillations with the Moon's Nodal Tide on the Northwest European Shelf (Eastern North Sea to Central Baltic Sea). Journal of Coastal Research, 31(5), 1041–1056

ABSTRACT Schmith, T.; Thejll, P., and Nielsen, J.W., 2016. Discussion of: Hansen, J. M., Aagaard, T. and Kuijpers, A., 2015. Sea-level forcing by synchronization of 56- and 74-year oscillations with the Moon's nodal tide on the northwest European shelf (eastern North Sea to central Baltic Sea). Journal of Coastal Research, 31(5), 1041–1056. We criticize important aspects of the analysis presented in Hansen, Aagaard, and Kuijpers (2015) (hereafter “HAK”) and thereby cast doubt on their conclusions. HAK claim that 18.6-year variations in sea level are supported by tidal theory, but this is not the case; therefore, the existence of such variations must be explicitly shown. We calculated the amplitude spectrum of the annual sea level by harmonic analysis and found no significant peaks at the periods claimed by HAK. Next, we used results given by HAK to reconstruct their decomposition and formed the residuals by subtracting the decomposition from the original data. We found that the variability near 18.6 years...

Atmospheric studies from the Mars Science Laboratory Entry, Descent and Landing atmospheric structure reconstruction

The Mars Science Laboratory (MSL) entered the martian atmosphere on Aug. 6, 2012 landing in Gale crater (4.6°S, 137.4°E) in the local mid-afternoon. Aerodynamic accelerations were measured during descent and atmospheric density, pressure and temperature profiles have been calculated from this data. Using an averaging technique developed for the NASA Phoenix Mars mission, the profiles are extended to 134.1km, twice that of the engineering reconstruction. Large-scale temperature oscillations in the MSL temperature profile are suggestive of thermal tides. Comparing the MSL temperature profile with measured Mars Climate Sounder temperature profiles and Mars Climate Database model output highlights the presence of diurnal tides. Derived vertical wavelengths for the diurnal migrating tide are larger than predicted from idealized tidal theory, indicating an added presence of nonmigrating diurnal tides. Sub-CO2 condensation mesospheric temperatures, very similar to the Pathfinder temperature profile, allude to the possibility of CO2 clouds. This is however not supported by recent observations and models.

TIDAL EVOLUTION OF ASTEROIDAL BINARIES. RULED BY VISCOSITY. IGNORANT OF RIGIDITY

The rate of tidal evolution of asteroidal binaries is defined by the dynamical Love numbers divided by quality factors. Common is the (often illegitimate) approximation of the dynamical Love numbers with their static counterparts. As the static Love numbers are, approximately, proportional to the inverse rigidity, this renders a popular fallacy that the tidal evolution rate is determined by the product of the rigidity by the quality factor: $\,k_l/Q\propto 1/(\mu Q)\,$. In reality, the dynamical Love numbers depend on the tidal frequency and all rheological parameters of the tidally perturbed body (not just rigidity). We demonstrate that in asteroidal binaries the rigidity of their components plays virtually no role in tidal friction and tidal lagging, and thereby has almost no influence on the intensity of tidal interactions (tidal torques, tidal dissipation, tidally induced changes of the orbit). A key quantity that determines the tidal evolution is a product of the effective viscosity $\,\eta\,$ by the tidal frequency $\,\chi\,$. The functional form of the torque's dependence on this product depends on who wins in the competition between viscosity and self-gravitation. Hence a quantitative criterion, to distinguish between two regimes. For higher values of $\,\eta\chi\,$ we get $\,k_l/Q\propto 1/(\eta\chi)\;$; $\,$while for lower values we obtain $\,k_l/Q\propto \eta\chi\,$. Our study rests on an assumption that asteroids can be treated as Maxwell bodies. Applicable to rigid rocks at low frequencies, this approximation is used here also for rubble piles, due to the lack of a better model. In the future, as we learn more about mechanics of granular mixtures in a weak gravity field, we may have to amend the tidal theory with other rheological parameters, ones that do not show up in the description of viscoelastic bodies.

INTERPLAY OF TIDAL EVOLUTION AND STELLAR WIND BRAKING IN THE ROTATION OF STARS HOSTING MASSIVE CLOSE-IN PLANETS

This paper deals with the application of the creep tide theory (Ferraz-Mello, Cel. Mech. Dyn. Astron. vol. 116, 109, 2013) to the study of the rotation of stars hosting massive close-in planets. The stars have nearly the same tidal relaxation factors as gaseous planets and the evolution of their rotation is similar to that of close-in hot Jupiters: they tidally evolve towards a stationary solution. However, stellar rotation may also be affected by stellar wind braking. Thus, while the rotation of a quiet host star evolves towards a stationary attractor with a frequency ($1+6e^2$) times the orbital mean-motion of the companion, the continuous loss of angular momentum in an active star displaces the stationary solution towards slower values: Active host stars with big close-in companions tend to have rotational periods larger than the orbital periods of their companions. The study of some hypothetical examples shows that because of tidal evolution, the rules of gyrochronology cannot be used to estimate the age of one system with a large close-in companion, no matter if the star is quiet or active, if the current semi-major axis of the companion is smaller than 0.03--0.04 AU. Details on the evolution of the systems: CoRoT LRc06E21637, CoRoT-27, Kepler-75, CoRoT-2, CoRoT-18, CoRoT-14 and on hypothetical systems with planets of mass 1--4 M_Jup in orbit around a star similar to the Sun are given.

The small and large lags of the elastic and anelastic tides

The aim of this letter is to discuss the virtual identity of two recent tidal theories: the creep tide theory of Ferraz-Mello (Cel. Mech. Dyn. Astron. 116, 109, 2013) and the Maxwell model developed by Correia et al. (Astron. Astrophys. 571, A50, 2014). It includes the discussion of the basic equations of the theories, which, in both cases, include an elastic and an anelastic component, and shows that the basic equations of the two theories are equivalent and differ by only a numerical factor in the anelastic tide. It also includes a discussion of the lags: the lag of the full tide (geodetic), dominated by the elastic component, and the phase of the anelastic tide. In rotating rocky bodies not trapped in a spin-orbit resonance (e.g., the Earth) the geodetic lag is close to zero and the phase of the semi-diurnal argument in the anelastic tide is close to 90 degrees. The results obtained from combining tidal solutions from satellite tracking data and from Topex/Poseidon satellite altimeter data, by Ray et al., are extended to determine the phase of the semi-diurnal argument in the Earth's anelastic tide as sigma_0=89.80 \pm 0.05 degrees.

Tidal synchronization of close-in satellites and exoplanets: II. Spin dynamics and extension to Mercury and exoplanet host stars

This paper deals with the application of the creep tide theory (Ferraz-Mello, CeMDA 116, 109, 2013) to the rotation of close-in satellites, Mercury, close-in exoplanets and their host stars. The solutions show two extreme cases: close-in giant gaseous planets, with fast relaxation (low viscosity) and satellites and Earth-like planets, with slow relaxation (high viscosity). The rotation of close-in gaseous planets follows the classical Darwinian pattern: it is tidally driven towards a stationary solution which is synchronized, but, if the orbit is elliptical, with a frequency larger than the orbital mean-motion. The rotation of rocky bodies, however, may be driven to several attractors whose frequencies are 1/2,1,3/2,2,5/2 ... times the mean-motion. The number of attractors increases with the viscosity of the body and with the orbital eccentricity. The classical example is Mercury, whose rotational period is 2/3 of the orbital period (3/2 attractor). The planet behaves as a molten body with a relaxation that allowed it to cross the 2/1 attractor without being trapped, but not to escape being trapped in the 3/2 one. In that case, the relaxation is estimated to lie in the interval 4.6 -- 27 x 10^{-9} s^{-1} (equivalent to a quality factor roughly constrained to the interval 5<Q<50). The stars have relaxation similar to the hot Jupiters and their rotation is also driven to the only stationary solution existing in these cases. However, solar-type stars may lose angular momentum due to stellar wind, braking the rotation and displacing the attractor towards larger periods. Old active host stars with big close-in companions generally have rotational periods larger than the orbital periods of the companions. The paper also includes the study of the energy dissipation and the evolution of the orbital eccentricity.

Galileo and Descartes on Copernicanism and the cause of the tides

Galileo and Descartes were on the front lines of the defense of Copernicanism against theological objections that took on special importance during the seventeenth century. Galileo attempted to overcome opposition to Copernicanism within the Catholic Church by offering a demonstration of this theory that appeals to the fact that the double motion of the earth is necessary as a cause of the tides. It turns out, however, that the details of Galileo's tidal theory compromise his demonstration. Far from attempting to provide a demonstration of the earth's motion, Descartes ultimately argued that his system is compatible with the determination of the Church that the earth is at rest. Nonetheless, Descartes's account of the cause of the tides creates difficulty for this argument.