On the tidal effects in the motion of artificial satellites

The general perturbations in the elliptic and vectorial elements of a satellite as caused by the tidal deformations of the non-spherical Earth are developed into trigonometric series in the standard ecliptical arguments of Hill-Brown lunar theory and in the equatorinal elements ω and Ω of the satellite. The integration of the differential equations for variation of elements of the satellite in this theory is easy because all arguments are linear or nearly linear in time. The trigonometrical expansion permits a judgment about the relative significance of the amplitudes and periods of different tidal ‘waves’ over a long period of time. Graphs are presented of the tidal perturbations in the elliptic elements of the BE-C satellite which illustrate long term periodic behavior. The tidal effects are clearly noticeable in the observations and their comparison with the theory permits improvement of the ‘global’ Love numbers for the Earth.     

Tidal parameters from the variation of inclination of GEOS-1 and GEOS-2

Analysis of the luni-solar tidal perturbations of the inclination of GEOS-1 (1965-89A,i=59°) and GEOS-2 (1968-002A,i=106°.) has yielded the valuesk ₂=0.22 (=0.02) and 0.31 (=0.01) respectively for the apparent second degree Love number. For GEOS-1 a new purely numerical method involvingosculating elements was employed. For GEOS-2 it was necessary to analyze the variations of themean elements because of the very long period (450 days) of the dominant solar tidal perturbation. An additional analysis of the variation of the mean elements of GEOS-1 confirmed the value ofk ₂ obtained from the osculating elements. The disparate values indicate that the simple 2nd degree zonal harmonic model of the tidal potential employed by ourselves and all other investigators is accommodating other effects in addition to those caused by the solid Earth tides. A recent paper by Lambecket al. (1973) indicates that ocean tide effects have significant perturbations on satellite orbits and cannot be neglected.     

Tidally induced surface displacements, external potential variations, and gravity variations on Mars

We have used and extended Roosbeek’s tidal potential for Mars to calculate tidal displacements, gravity variations, and external gravitational potential variations. The tides on Mars are caused by the Sun, and to a lesser degree by the natural satellites Phobos (8%, relative to the Sun) and Deimos (0.08%, relative to the Sun). To determine the reaction of Mars to the tidal forcing, the Love numbers h, l, and k and the gravimetric factor δ were calculated for interior models of Mars with different state, density, and radius of the core and for models which include mantle anelasticity. The latitude dependence and frequency dependence of the Love numbers have been taken explicitly into account. The Love numbers are about three times smaller than those for the Earth and are very sensitive to core changes; e.g., a difference of about 30% is found between a model with a liquid core and an otherwise similar model with a solid core. Tidal displacements on Mars are much smaller than on Earth due to the smaller tidal potential, but also due to the smaller reaction of Mars (smaller Love numbers). For both the tidal diplacement and the tidal external potential perturbations, the tidal signal is at the limit of detection and is too small to permit properties of Mars’s interior to be inferred. On the other hand, the Phobos tidally induced gravity changes, which are subdiurnal with typical periods shorter than 12 h, can be measured very precisely by the very broad band seismometer with thermal control of the seismological experiment SEIS of the upcoming NetLander mission. It is shown that the Phobos-induced gravity tides could be used to study the Martian core.     

Exoplanet's Figure and Its Interior

Along with the development of the observing technology, the observation and study on the exoplanets’ oblateness and apsidal precession have achieved significant progress. The oblateness of an exoplanet is determined by its interior density profile and rotation period. Between its Love number k₂ and core size exists obviously a negative correlation. So oblateness and k₂ can well constrain its interior structure. Starting from the Lane-Emden equation, the planet models based on different polytropic indices are built. Then the flattening factors are obtained by solving the Wavre's integro-differential equation. The result shows that the smaller the polytropic index, the faster the rotation, and the larger the oblateness. We have selected 469 exoplanets, which have simultaneously the observed or estimated values of radius, mass, and orbit period from the NASA (National Aeronautics and Space Administration) Exoplanet Archive, and calculated their flattening factors under the two assumptions: tidal locking and fixed rotation period of 10.55 hours. The result shows that the flattening factors are too small to be detected under the tidal locking assumption, and that 28% of exoplanets have the flattening factors larger than 0.1 under the fixed rotation period of 10.55 hours. The Love numbers under the different polytropic models are solved by the Zharkov's approach, and the relation between k₂ and core size is discussed.     

Ocean tide perturbations in the orbits of GEOS 1 and GEOS 2

Douglaset al. (1973) have estimated tidal parameters from the orbits of GEOS 1 and GEOS 2. Their results, interpreted in terms of Love numbers, are rather dispersive due in part to their neglect of the ocean tides. The ocean tidal corrections are estimated in this paper, but although they do not explain all of the discrepancy they do emphasise the importance of these perturbations on the motion of close Earth satellites. The remaining discrepancies could result in part from the fact that part of the long period tidal perturbations have been absorbed by the zonal harmonics in the Earth's gravity field.     

On the stroboscopic method of second order

It is assumed that the dynamical system can be represented by equations of the form $$\begin{gathered} \dot x = \varepsilon _i f_i (x,y) \hfill \\ \dot y = u(x,y) + \varepsilon _i g_i (x,y) \hfill \\ \end{gathered} $$ as this is the case for the Lagrange equations in celestial mechanics. The perturbation functionsf _i andg _i may also depend on the timet. The fast angular variabley is now taken as independent variable. Using perturbation theory and expanding in Taylor series the differential equations for the zeroth, first, second, ... order approximations are obtained. In the stroboscopic method in particular the integration is performed analytically over one revolution, say from perigee to perigee. By the rectification step applied tox andt, the initial values for the next revolution are obtained. It is shown how the second order terms can be determined for the various perturbations occurring in satellite theory. The solution constructed in this way remains valid for thousands of revolutions. An important feature of the method is the small amount of computing time needed compared with numerical integration.     

Fast computation of high eccentricity orbits by the stroboscopic method

In order to compute with a reasonable accuracy satellite orbits subjected to perturbations byJ ₂, the Sun, the Moon and airdrag, a first order expansion is used. The Lagrange equations are solved semi-analytically by the stroboscopic method. The intermediate-, long-periodic and secular terms are obtained, but if desired the same formulism also produces the short-periodic terms. The method is well suited for use on a computer and requires only about 1% of the computing time needed for numerical integration.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

Moments of inertia and the Chandlerian period for two-and three-layer models of the Galilean satellite Io

We consider two-layer (Fe-FeS core+silicate mantle) and three-layer (Fe-FeS core+silicate mantle+crust) models of the Galilean satellite Io. Two parameters are known from observations for the equilibrium figure of the satellite, the mean density ρ₀ and the Love number k₂. Previously, the Radau-Darwin formula was used to determine the mean moment of inertia. Using formulas of the Figure Theory, we calculated the principal moments of inertia A, B, and C and the mean moment of inertia I for the two-and three-layer models of Io using ρ₀ and k₂ as the boundary conditions. We concluded that when modeling the internal structure of Io, it is better to use the observed value of k₂ than the moment of inertia I derived from k₂ using the Radau-Darwin formula. For the models under consideration, we calculated the Chandlerian wobble periods of Io. For the three-layer model, this period is approximately 460 days.     

Can Cassini detect a subsurface ocean in Titan from gravity measurements?

Recent models of Titan's interior predict that the satellite contains an ocean of water and ammonia under an icy layer. Direct evidence for the presence of an ocean can be provided on the Cassini mission only by radio science determination of Titan Love number k₂. Simulations that use the five flybys T11, T22 T33, T45, and T68 (the latter two belonging to the extended mission) lead to the result that in the elastic case, where the Love number is real, k₂ will be determined with a one-sigma accuracy of 0.1. In the viscoelastic case, where k₂ is complex, the real and imaginary parts of k₂ will be determined with one sigma accuracies of 0.138 and 0.115, respectively. Ocean and oceanless models that include a viscoelastic rheology are built. In the viscoelastic case, there is a 93% probability to correctly predict the presence or absence of an ocean; this probability improves to 97% in the elastic case.     

Love numbers of the moon and of the terrestrial planets

In the IERS Standards (1989), for the Moon the adopted value of the tide Love number, k ₂, is equal to 0.0222. In this paper using the latest geodetic parameters of the Moon a group of internal structure models are constructed for this celestial body (see Table V), then the dependence of the Moon's core size on calculated value of k ₂ is explored. The obtained results indicate that the second degree Love number, k ₂ = 0.02664, of the lunar model 91–04 is near its observed value (0.027 ± 0.006). This implies that the Moon may possess an outer core of 660 km radius and of 300 kbar mean rigidity. With the same method the static Love numbers from degree 2 to 30 are computed for the terrestrial planets — Mercury, Venus, and Mars (see Table VII), and the influence of some parameters (such as the rigidity) of the outer core on low degree Love numbers is discussed. Finally, the likely range of the second degree Love numbers is determined for the terrestrial planets (see Table XI). It seems that if low degree Love numbers of a terrestrial planet can be detected in the future space explorations, there is some possibility to improve the planetary internal structure model. For example, as soon as space techniques yield an observed value of k ₂ > 0.10 for Mercury, there will be reason to anticipate that a partly melted iron core exists in this planet.     

Numerical theory of rotation of the deformable Earth with the two-layer fluid core. Part 1: Mathematical model

Improved differential equations of the rotation of the deformable Earth with the two-layer fluid core are developed. The equations describe both the precession-nutational motion and the axial rotation (i.e. variations of the Universal Time UT). Poincaré’s method of modeling the dynamical effects of the fluid core, and Sasao’s approach for calculating the tidal interaction between the core and mantle in terms of the dynamical Love number are generalized for the case of the two-layer fluid core. Some important perturbations ignored in the currently adopted theory of the Earth’s rotation are considered. In particular, these are the perturbing torques induced by redistribution of the density within the Earth due to the tidal deformations of the Earth and its core (including the effects of the dissipative cross interaction of the lunar tides with the Sun and the solar tides with the Moon). Perturbations of this kind could not be accounted for in the adopted Nutation IAU 2000, in which the tidal variations of the moments of inertia of the mantle and core are the only body tide effects taken into consideration. The equations explicitly depend on the three tidal phase lags δ, δ _c, δ _i responsible for dissipation of energy in the Earth as a whole, and in its external and inner cores, respectively. Apart from the tidal effects, the differential equations account for the non-tidal interaction between the mantle and external core near their boundary. The equations are presented in a simple close form suitable for numerical integration. Such integration has been carried out with subsequent fitting the constructed numerical theory to the VLBI-based Celestial Pole positions and variations of UT for the time span 1984–2005. Details of the fitting are given in the second part of this work presented as a separate paper (Krasinsky and Vasilyev 2006) hereafter referred to as Paper 2. The resulting Weighted Root Mean Square (WRMS) errors of the residuals dθ, sin θd for the angles of nutation θ and precession are 0.136 mas and 0.129 mas, respectively. They are significantly less than the corresponding values 0.172 and 0.165 mas for IAU 2000 theory. The WRMS error of the UT residuals is 18 ms.     

Ocean tidal perturbations on the orbit of the satellite “STARLETTE”

A continuous 300-day, world-wide laser-ranging data of the satellite STARLETTE between October 1976 and July 1977 was used to analyse the perturbation of the orbit by ocean tide. The long-period (longer than 20d) perturbations in the orbital inclination by the four main tide components K₁, P₁, K₂, S₂ were clearly identifiable in the maximum entropy power spectrum and periodogram. By comparing orbits including and excluding the tidal terms, the effect of the tide on the position of the satellite over 5 days was found to be of the order of one meter.     

Modeling stresses on satellites due to nonsynchronous rotation and orbital eccentricity using gravitational potential theory

The tidal stress at the surface of a satellite is derived from the gravitational potential of the satellite's parent planet, assuming that the satellite is fully differentiated into a silicate core, a global subsurface ocean, and a decoupled, viscoelastic lithospheric shell. We consider two types of time variability for the tidal force acting on the shell: one caused by the satellite's eccentric orbit within the planet's gravitational field (diurnal tides), and one due to nonsynchronous rotation (NSR) of the shell relative to the satellite's core, which is presumed to be tidally locked. In calculating surface stresses, this method allows the Love numbers h and ?, describing the satellite's tidal response, to be specified independently; it allows the use of frequency-dependent viscoelastic rheologies (e.g. a Maxwell solid); and its mathematical form is amenable to the inclusion of stresses due to individual tides. The lithosphere can respond to NSR forcing either viscously or elastically depending on the value of the parameter , where μ and η are the shear modulus and viscosity of the shell respectively, and ω is the NSR forcing frequency. Δ is proportional to the ratio of the forcing period to the viscous relaxation time. When Δ?1 the response is nearly fluid; when Δ?1 it is nearly elastic. In the elastic case, tensile stresses due to NSR on Europa can be as large as ∼3.3 MPa, which dominate the ∼50 kPa stresses predicted to result from Europa's diurnal tides. The faster the viscous relaxation the smaller the NSR stresses, such that diurnal stresses dominate when Δ?100. Given the uncertainty in current estimates of the NSR period and of the viscosity of Europa's ice shell, it is unclear which tide should be dominant. For Europa, tidal stresses are relatively insensitive both to the rheological structure beneath the ice layer and to the thickness of the icy shell. The phase shift between the tidal potential and the resulting stresses increases with Δ. This shift can displace the NSR stresses longitudinally by as much as 45° in the direction opposite of the satellite's rotation.     

Numerical integration of periodic orbits in the main problem of artificial satellite theory

We describe a collection of results obtained by numerical integration of orbits in the main problem of artificial satellite theory (theJ ₂ problem). The periodic orbits have been classified according to their stability and the Poincaré surfaces of section computed for different values ofJ ₂ andH (whereH is thez-component of angular momentum). The problem was scaled down to a fixed value (–1/2) of the energy constant. It is found that the pseudo-circular periodic solution plays a fundamental role. They are the equivalent of the Poincaré first-kind solutions in the three-body problem. The integration of the variational equations shows that these pseudo-circular solutions are stable, except in a very narrow band near the critical inclincation. This results in a sequence of bifurcations near the critical inclination, refining therefore some known results on the critical inclination, for instance by Izsak (1963), Jupp (1975, 1980) and Cushman (1983). We also verify that the double pitchfork bifurcation around the critical inclination exists for large values ofJ ₂, as large as |J ₂|=0.2. Other secondary (higher-order) bifurcations are also described. The equations of motion were integrated in rotating meridian coordinates.     

Simultaneous determination of global topography, tidal Love number and libration amplitude of Mercury by laser altimetry

Solar tidal forces generate elevation changes of Mercury's surface of the order 1 m within one Hermean year, and solar torques on the non-symmetric permanent mass distribution of the planet cause an uneven rotation of Mercury's surface with a libration amplitude of the order of 40 arcsec. Knowledge of the precise reaction of the planet to tidal forcing, expressed by the Love numbers h₂ and k₂, as well as accurate knowledge of the amplitude of forced libration Φ_lib, puts constraints on the internal structure, for example the state and the size of the core. The MESSENGER and BepiColombo missions to Mercury carry laser altimeters, whose primary goal is to accurately map the topography. Here we investigate if the Love number h₂ and the amplitude of forced libration can be determined together with the static topography of the planet from a global altimetry record. We do this by creating synthetic altimeter data for the nominal orbit of BepiColombo over the nominal mission duration of approximately four Mercury years and inverting them for the static and time-dependent parts of the topography. We assume purely Gaussian noise. We find that it is possible to extract both parameters h₂ and Φ_lib with an accuracy of approximately 10%, while the static topography coefficients of a spherical harmonic expansion can be determined simultaneously with an accuracy at the centimetre level. Extraction of the static topography to higher harmonic degrees improves the precision of the measurement of h₂ and Φ_lib. The simulation results demonstrate that it seems feasible to test current models on Mercury's interior with sufficient precision using BepiColombo Laser Altimeter data.     

Dynamical History of the Earth–Moon System

Differential equations describing the tidal evolution of the earth's rotation and of the lunar orbital motion are presented in a simple close form. The equations differ in form for orbits fixed to the terrestrial equator and for orbits with the nodes precessing along the ecliptic due to solar perturbations. Analytical considerations show that if the contemporary lunar orbit were equatorial the evolution would develop from an unstable geosynchronous orbit of the period about 4.42 h (in the past) to a stable geosynchronous orbit of the period about 44.8 days (in the future). It is also demonstrated that at the contemporary epoch the orbital plane of the fictitious equatorial moon would be unstable in the Liapunov's sense, being asymptotically stable at early stages of the evolution. Evolution of the currently near-ecliptical lunar orbit and of the terrestrial rotation is traced backward in time by numerical integration of the evolutional equations. It is confirmed that about 1.8 billion years ago a critical phase of the evolution took place when the equatorial inclination of the moon reached small values and the moon was in a near vicinity of the earth. Before the critical epoch t _cr two types of the evolution are possible, which at present cannot be unambiguously distinguished with the help of the purely dynamical considerations. In the scenario that seems to be the most realistic from the physical point of view, the evolution also has started from a geosynchronous equatorial lunar orbit of the period 4.19 h. At t < t _cr the lunar orbit has been fixed to the precessing terrestrial equator by strong perturbations from the earth's flattening and by tidal effects; at the critical epoch the solar perturbations begin to dominate and transfer the moon to its contemporary near-ecliptical orbit which evolves now to the stable geosynchronous state. Probably this scenario is in favour of the Darwin's hypothesis about originating the moon by its separation from the earth. Too much short time scale of the evolution in this model might be enlarged if the dissipative Q factor had somewhat larger values in the past than in the present epoch. Values of the length of day and the length of month, estimated from paleontological data, are confronted with the results of the developed model.     

Perturbations of a close-earth satellite due to sunlight diffusely reflected from the Earth

A semianalytical method has been developed to calculate the radiation-pressure perturbations of a close-Earth satellite due to sunlight reflected from the Earth. It is assumed that the satellite is spherically symmetric and that the solar radiation is reflected from the Earth according to Lambert's Law. To account for the increasing reflectivity of the Earth toward the poles, its albedo is assumed to have a latitudinal dependence given bya=a ₀ +a ₂ sin². The effect of the terminator on the perturbations has been neglected. The perturbations within a particular revolution are given analytically, while the long-range perturbations are obtained by accumulation.     

The lunar orbit revisited,III

In this paper we present an investigation on the tidal evolution of a system of three bodies: the Earth, the Moon and the Sun. Equations are derived including dissipation in the planet caused by the tidal interaction between the planet and the satellite and between the planet and the sun. Dissipation within the Moon is included as well. The set of differential equations obtained is valid as long as the solar disturbances dominate the perturbations on the satellite's motion due to the oblateness of the planet, namelya/R _e greater than 15, and closer than that point equations derived in a preceding paper are used.The result shows the Moon was closer to the Earth in the past than now with an inclination to the ecliptic greater than today, whereas the obliquity was smaller. Toward the past, the inclination to the Earth's equator begins decreasing to 12° fora/R _e=12 and suddenly grows. During the first stage the results are weakly dependant on the magnitude of the dissipation within the satellite, whereas the distance of the closest approach and the prior history are strongly dependent on that dissipation. In particular, the crossing of the Roche limit can be avoided.     

Periodic perturbations in close binary systems due to tidal lag

In two previous papers (Zafiropoulos and Kopal, 1983a, b; hereafter referred to as Papers I and II) we have investigated the effects of rotational and tidal distortion (for non-lagging tides) on the orbital elements of a close binary system. The present paper deals with secular and periodic perturbations caused by dynamical tides. The componentsR, S, andW of disturbing accelerations for tidal lag have been substituted in the Gaussian form of Lagrange's planetary equations to give the first-order approximation. The results obtained have been expressed by means of Hansen coefficients and include the effects produced by the second, third and fourth harmonic dynamical tides.