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NASA NTRS1973

Some effects of elasticity on lunar rotation

A general Hamiltonian for a rotating moon in the field of the earth is expanded in terms of parameters orienting the spin angular momentum relative to the principal axes of the moon and relative to coordinate axes fixed in the orbital plane. The effects of elastic distortion are included as modifications of the moment of inertia tensor, where the magnitude of the distortion is parameterized by the Love number k2. The amplitudes of the principal periodic terms in the longitude of a point on the moon due to variations of the tide caused by the earth are discussed. A detailed derivation of the free wobble of the lunar spin axis about the axis of maximum moment of inertia is given. If the moon's rigidity is close to that of the earth there is no effect of elasticity on the rotation which can be measured with the laser ranging and, therefore, no elastic properties of the moon can be determined from variations in the rotation.

Peale, S. J.↗
NASA NTRS1974

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 and GEOS-2 has yielded the values 0.22 and 0.31 respectively for the apparent second degree Love number. For GEOS-1 a new purely numerical method involving osculating elements was employed. For GEOS-2 it was necessary to analyze the variations of the mean elements because of the very long period (450 days) of the dominant solar tidal perturbation. The disparate values indicate that the simple second degree zonal harmonic model of the tidal potential is accommodating other effects in addition to those caused by the solid earth tides. A recent paper by Lambeck et al. (1973) indicates that ocean tide effects have significant perturbations on satellite orbits and cannot be neglected.

Douglas, B. C.↗
NASA NTRS1976

Analyses of the solid earth and ocean tidal perturbations on the orbits of the Geos 1 and Geos 2 satellites

Perturbations in the inclination of the Geos 1 and Geos 2 satellite orbits have been analyzed for the solid earth and ocean tide contributions. Precision reduced camera and Tranet Doppler observations spanning periods of over 600 days for each satellite were used to derive mean orbital elements. Perturbations due to the earth's gravity field, solar radiation pressure, and atmospheric drag were modeled, and the resulting inclination residuals were analyzed for tidal effects. The amplitudes of the observed total tidal effects were about 1.2 arc sec (36 m) in the inclination of Geos 1 and 4.5 arc sec (135 m) for Geos 2. The solid earth tides were then modeled by using the Love number 0.30. The resulting inclination residuals were then analyzed for ocean tide spherical harmonic parameters.

Felsentreger, T. L.↗
NASA NTRS1977

Tidal perturbations on the satellite 1967-92A

The orbit of the 1967-92A satellite was studied to ascertain the extent to which tidal forces contribute to orbital perturbations. Parameters describing the ocean tide potential-in particular for the M2 and S2 constituents-were estimated. Since the ocean tide potential is less well known than the solid Earth tide, the ocean tide parameter estimation is based upon the use of a value of 0.3 for the solid Earth tide Love number in the orbit determination procedure. These tidal parameter values are in good agreement with those appearing in numerical models of the M2 and S2 tides derived from surface data.

Felsentreger, T. L.↗
NASA NTRS1975

Geodetic and Astrometric Measurements with Very-Long-Baseline Interferometry

The use of very-long-baseline interferometry (VLBI) observations for the estimation of geodetic and astrometric parameters is discussed. Analytic models for the dependence of delay and delay rate on these parameters are developed and used for parameter estimation by the method of weighted least squares. Results are presented from approximately 15,000 delay and delay-rate observations, obtained in a series of nineteen VLBI experiments involving a total of five stations on two continents. The closure of baseline triangles is investigated and found to be consistent with the scatter of the various baseline-component results. Estimates are made of the wobble of the earth's pole and of the irregularities in the earth's rotation rate. Estimates are also made of the precession constant and of the vertical Love number, for which a value of 0.55 + or - 0.05 was obtained.

Robertson, D. S.↗
NASA NTRS1978

Tidal perturbations on the satellite 1967-92A

The orbit of the 1967-92A satellite has been studied to ascertain the extent to which tidal forces contribute to orbital perturbations. This study has permitted an estimation of the magnitudes of ocean tide effects on the satellite's inclination, in particular for the M sub 2 and S sub 2 constituents. The ocean tide estimates are based upon the use of a value of 0.3 for the solid earth tide Love number and a lag angle of zero deg in the orbit determination procedure. The amplitudes and phases of these tidal effects are in good agreement with those calculated from numerical models of the tidal parameters derived from surface data.

Felsentreger, T. L.↗
NASA NTRS1978

Lunar tidal acceleration obtained from satellite-derived ocean tide parameters

One hundred sets of mean elements of GEOS-3 computed at 2-day intervals yielded observation equations for the M sub 2 ocean tide from the long periodic variations of the inclination and node of the orbit. The 2nd degree Love number was given the value k sub 2 = 0.30 and the solid tide phase angle was taken to be zero. Combining obtained equations with results for the satellite 1967-92A gives the M sub 2 ocean tide parameter values. Under the same assumption of zero solid tide phase lag, the lunar tidal acceleration was found mostly due to the C sub 22 term in the expansion of the M sub 2 tide with additional small contributions from the 0 sub 1 and N sub 2 tides. Using Lambeck's (1975) estimates for the latter, the obtained acceleration in lunar longitudal in excellent agreement with the most recent determinations from ancient and modern astronomical data.

Goad, C. C.↗
NASA NTRS1978

Tidal friction in the solid earth

The earth's imperfectly elastic response to body and loading tidal forces is discussed using complex Love numbers and complex mass loading coefficients. Exact analytical expressions were derived relating the energy dissipation within an inhomogeneous, compressible solid earth to the surface values of these complex characteristic numbers, thus relating the global dissipation function Q to the phase shifts in the potential, gravity, tilt, strain and displacement tides. Integration of a global ocean tidal model shows that energy dissipated in the solid earth due to ocean loading is at least 10% of that dissipated in the body tides.

Zschau, J.↗
NASA NTRS1978

Body tides on an elliptical rotating earth

The complete tidal response of an elliptical, rotating, elastic Earth is found to contain small displacements which do not fit into the conventional Love number framework. Corresponding observable tidal quantities (gravity, tilt, strain, Eulerian potential, etc.) are modified by the addition of small latitude dependent terms.

Wahr, J. M.↗
NASA NTRS1979

Investigation of tidal displacements of the earth's surface by laser ranging to GEOS-3

Two different approaches to the problem were investigated: (1) a dynamic method that computed pass-to-pass apparent movements in station height relative to short arcs fitted to several passes of data from the same station, and (2) a quasi-geometric method that reduces the dependence on unmodelled satellite dynamics to a knowledge of only the radial position of the satellite by considering two-station simultaneous ranging at the precise time that the satellite passes through the plane defined by the two stations and the center of the mass of the earth. The dynamic method was applied to 24-hour arcs in an attempt to study the total tidal displacement at Goddard, but it was found to be unsuitable. The quasi-geometry method, on the other hand, is significantly less influenced by these effects because it is sensitive principally to the differential tidal displacements between laser stations. Using this method, a good approximation to the theoretical Love number h sub 2 was obtained from five passes over the GEOS-3 calibration area when a systematic bias of 2.16 meters was allowed in the radial distance to the satellite.

Bower, D. R.↗
NASA NTRS1980

Crustal deformation at very long baseline interferometry sites due to seasonal air-mass and ground water variations

The seasonal deformation normal to the Earth's surface was calculated at stations involved or interested in very long baseline interferometry (VLBI) geodesy and at hypothetical sites in Australia and Brazil using global atmospheric pressure data, values for groundwater storage, and load Love numbers deduced from current Earth models. It was found that the annual range of deformation approached the centimeter level measuring potential of the VLBI technqiue at Greenbank, Haystack, and the Brazil site.

Stolz, A.↗
NASA NTRS1981

Tidal variations of earth rotation

The periodic variations of the earths' rotation resulting from the tidal deformation of the earth by the sun and moon were rederived including terms with amplitudes of 0.002 millisec and greater. The series applies to the mantle, crust, and oceans which rotate together for characteristic tidal periods; the scaling parameter is the ratio of the fraction of the Love number producing tidal variations in the moment of inertia of the coupled mantle and oceans (k) to the dimensionless polar moment of inertia of the coupled moments (C). The lunar laser ranging data shows that k/C at monthly and fortnightly frequencies equals 0.99 + or - 0.15 and 0.99 + or - 0.20 as compared to the theoretical value of 0.94 + or - 0.04.

Yoder, C. F.↗
NASA NTRS1982

Determination of intercontinental baselines and Earth orientation using VLBI

A series of experiments was conducted during the last decade to explore the capability of very long baseline interferometry (VLBI) to measure the crustal and rotational motions of the Earth with accuracies at the centimeter level. The observing stations are those of NASA's Deep Space Network in California, Spain and Australia. A multiparameter fit to the observed values of delay and delay rate yields radio source positions, polar motion, universal time, the precession constant, baseline vectors, and solid Earth tides. Source positions are obtained with formal errors of the order of 0''.01. UT1-UTC and polar motion are determined at 49 epochs, with formal error estimates for the more recent data of 0.5 msec for UT1-UTC and 2 to 6 mas for polar motion. Intercontinental baseline lengths are determined with formal errors of 5 to 10 cm. The Love numbers and Earth tide phase lag agree with the commonly accepted values.

Sovers, O. J.↗
NASA NTRS1983

Planetary geodesy

The current known geodetic parameters of the planets and their moons are reviewed. A 3:2 spin-orbit resonance has been calculated for Mercury, a planet for which previous mass estimates are suggested to be inaccurate by 30 pct. The Pioneer Venus Orbiter data indicated a Venus diameter of 6051 km, a geoid surface highly-correlated with the surface topography, and regional gravity anomalies. Reflective mirrors on the surface and data from the Lunar Orbiter 4 have produced values for the principal polar moment of inertia homogeneity factor, the earth-moon mass ratio, a fifth degree and order spherical harmonic gravity model, the lunar acceleration, and the lunar potential Love number. The Mars gravity field is now estimated up to 12th degree and order spherical harmonics from Mariner 9 and Viking Orbiters 1 and 2, which also obtained new topographical data. New gravity fields and mass are being calculated for the Jovian and Saturnian moons. The outer planets' radii and rotational periods have recently been revised to greater accuracy. New data will be forthcoming from the Space Telescope and the planned Venus Radar Mapper.

Sjogren, W. L.↗
NASA NTRS1984

The Laplacian resonance and tidal dissipation

The Laplacian resonance amongst Io, Europa, and Ganymede was examined, taking into account tidal dissipation in Jupiter, Io and Europa. In equilibrium, it is not possible to neglect dissipation in Europa nor the torques of Jupiter on Europa and Ganymede. A formal calculation was made on the assumption that the tidal torques and the torques in orbit-orbit resonance reached an equilibrium such that the rates of decrease of the mean motions of the satellites are in the ratios 4:2:1. Q(j)/k(j)=167 Q(1)/k(1), Q(2)/k(2) = 0.44 Q(1)/K(1), where Q denotes the quality at the body's frequency of rotation, k its second degree Love number and the subscripts J, 1 and 2 denote Jupiter, Io, and Europa. The Q(J) found is at Jupiter's frequency of rotation. The rate of tidal dissipation in Europa comes to 1/5 that of Io. Such a low value of Q(2)/K(2) is plausible if dissipation in both Io and Europa is due to tidal friction in fluid against underlying and overlying solid layers.

Cook, A. F.↗
NASA NTRS1984

Geoid anomalies in a dynamic earth

Potential and surface deformation Love numbers for internal loading have been calculated in order to obtain a dynamically consistent relationship between the geoid and the earth's response to internal buoyancy forces. These quantities depend on the depth and harmonic degree of loading, and can be integrated as Green functions to obtain the dynamic response due to an arbitrary distribution of internal density contrasts. Constructing a series of spherically symmetric, self-gravitating flow models for a variety of radial Newtonian viscosity variations and flow configurations, and calculating relaxation times for spherically symmetric viscous earth models, it is demonstrated that boundary deformation due to internal loading reaches its equilibrium value on the same time scale as postglacial rebound; this is much less time than the time scale for significant change in the convective flow pattern.

Richards, M. A.↗
NASA NTRS1984

Secular rotational motions and the mechanical structure of a dynamical viscoelastic earth

A survey is presented of analytical methods for computing the linear responses of the rotational axis of a layered viscoelastic earth to surface loading. Theoretical research in this area is first summarized, and the differences between the mechanical boundary conditions to be applied at the interface separating the upper and lower mantles for an adiabatically and chemically stratified mantle are discussed. Some examples of polar wander and secular variation of the spin rate from glacial excitation are presented for various types of chemical and viscosity stratifications. The effects of an artificial density jump at the base of the lithosphere in models are examined, and certain issues concerning the fluid tidal Love number for different types of density stratification are addressed. The meaning of effective plate thickness over geological time scales for rotational dynamics is discussed.

Yuen, D. A.↗
NASA NTRS1986

Geodesy by radio interferometry - Determinations of baseline vector, earth rotation, and solid earth tide parameters with the Mark I very long baseline radio interferometery system

Thirty-seven very long baseline radio interferometry experiments performed between 1972 and 1978 are analyzed and estimates of baseline vectors between six sites, five in the continental United States and one in Europe are derived. No evidence of significant changes in baseline length is found. For example, with a statistical level of confidence of approximately 85 percent, upper bounds on such changes within the United States ranged from a low of 10 mm/yr for the 850 km baseline between Westford, Massachusetts, and Green Bank, West Virginia, to a high of 90 mm/yr for the nearly 4000 km baseline between Westford and Goldstone, California. Estimates for universal time and for the x component of the position of the earth's pole are obtained. For the last 15 experiments, the only ones employing wideband receivers, the root-mean-square differences between the derived values and the corresponding ones published by the Bureau International de l'Heure are 0.0012 s and 0.018 arc sec respectively. The average value obtained for the radial Love number for the solid earth is 0.62 + or - 0.02 (estimated standard error).

Ryan, J. W.↗