Elliptical Earth Research Articles

Determination of the Earth’s free core nutation parameters by using tidal gravity data

The Earth’s free core nutation (FCN) is a retrograde eigenmode which is attributed to the interaction between the solid mantle and the liquid core of the rotational elliptical Earth. This mode appears as an eigenmode of nearly diurnal free wobble (NDFW) in a terrestrial reference frame with a period of about one day (XU et al, 2001). Therefore, the NDFW will lead to an obvious resonance enhancement in the diurnal tidal gravity observations, especially those of the tidal waves with frequencies closed to its eigenfrequency such as P1, K1, Ψ1 and Φ1. The FCN resonance parameters can be retrieved accurately by high-precision tidal gravity observations, especially those recorded with the superconducting gravimeters (SG). The Global Geodynamics Project (GGP) organized by IUGG took it as an important content for determining the FCN resonance parameters by using gravity data. However, the results are affected by many factors such as station location, background noise, the selection of the tide-generating potential tables, ocean tide models, data processing techniques and so on. In our study, the FCN parameters will be retrieved by using the SG observations at Wuhan, and the effects of the choices of various tide-generating potential tables, oceanic models and weight functions on the estimation of the FCN parameters will be discussed in detail.

Geographical variations of the 0S0 normal mode amplitude: predictions and observations after the Sumatra-Andaman earthquake

Abstract The radial seismic normal mode 0S0 was strongly excited by the 2004 Mw = 9.3 Sumatra-Andaman earthquake at a period of 20.5 min. In a spherically symmetric Earth model, 0S0 amplitude is the same everywhere on the Earth’s surface. However, when the ellipticity and rotation of the Earth are taken into consideration, theoretical computations predict an amplitude of 0S0 1% higher at the pole than at the equator. Based on a realistic three-dimensional heterogeneous rotating elliptic Earth model, our predictions indicate that the amplitude of 0S0 is 2% higher at the pole than at the equator. A longitude dependency of 0S0 amplitude is also shown. The analysis of 13 superconducting gravimeter (SG) records of the 2004 Sumatra-Andaman earthquake supports the predicted geographical variations of 0S0 amplitude. We have also obtained new estimates for the frequency and Q of 0S0: 0.8146566±1.6 10−6 mHz and 5506±19.

Geometrical Variations of Gain Patterns

This paper studies the effect of the altitude and the elliptical nature of the Earth on projected antenna pattern functions (APFs). These variations can play a significant role in spatial filter algorithms, such as the Backus-Gilbert method. In this paper, a solution for projecting an APF from an arbitrary sensor and platform onto an elliptical Earth is presented. A general time integration of this projection is presented, which is important for dynamically generating the effective antenna pattern functions (EAPFs) in cases where the variations are nonnegligible. It is shown that the elliptical nature of the Earth provides a variation of order 1% in the area of the EAPF. These results suggest that approximating the Earth as a sphere is valid. It is also shown that elliptical orbits should consider the implication of the altitude variation before precomputing EAPFs

The explicit scalar equations of infinitesimal elastic-gravitational motion in the rotating, slightly elliptical fluid outer core of the Earth

Smith (1974) derived a set of scalar equations of infinitesimal elastic-gravitational motion for a rotating, slightly elliptical Earth. In the fluid outer core (FOC), the tangential tractions vanish identically and the differential equations for the transverse displacement fields do not hold any longer, but the derivative of the transverse displacements are needed in other differential equations, it is therefore the purpose of this paper to get the derivative of the transverse displacements as well as other differential equations in an explicit and ready-to-program format. Our derivation is shown to be more direct and explicit than that suggested by Smith (1974).

Modeling and Simulation of Aerospace Vehicle Dynamics

11R9. Modeling and Simulation of Aerospace Vehicle Dynamics. AIAA Education Series. - PH Zipfel (Univ of Florida, Gainesville FL). AIAA, Reston VA. 2000. 551 pp. ISBN 1-56347-456-5. $79.95.Reviewed by W Schiehlen (Inst B of Mech, Univ of Stuttgart, Pfaffenwaldring 9, Stuttgart, 70550, Germany).This book can be used for one- and two-semester courses as well as for self-study. It presents the fundamentals of aerospace vehicle dynamics, and for simulations, the CADAC environment is available free of charge from the AIAA home page. Both components result in an excellent learning environment. The book starts with an overview of today’s computed-aided engineering concepts also called virtual engineering. Then, the two parts of the book Modeling of Flight Dynamics with six chapters and Simulation of Aerospace Vehicles with four chapters, are introduced. In the second chapter, the mathematical concepts of modeling are discussed based on classical mechanics and tensor calculus. Most important for aerospace vehicles are frames and coordinate systems which are presented in Chapter 3. The kinematics of translation and rotation are treated in Chapter 4 using the author’s concept of the rotational time derivative which has nice invariance properties. Then, in Chapter 5 the translational dynamics are considered based on Newtonian Dynamics and the corresponding transformations in moving reference frames. The rotational dynamics presented in Chapter 6 use Euler’s Law and includes also some aspects of gyrodynamics. The full 3D equations of motion are complex and not so easy to understand. In a first step, the perturbation techniques are introduced in Chapter 7 to get some insight in the linear behavior of aerospace vehicles. However, the results are restricted particular motions. Therefore, it is quite natural to proceed to the second part of the book. Chapter 8 is devoted to three-degree-of-freedom simulations with subsystem models for the atmosphere, the gravitational attraction, the drag, and the propulsion. Five-degree-of-freedom simulations follow in Chapter 9 with extended subsystems and more detailed missile simulations. Finally, in Chapter 10, six-degree-of-freedom simulations are introduced for flat Earth and elliptic Earth equations of motion. Subsystems for aerodynamics, autopilots, actuators, inertial navigation, guidance, and infrared sensors are added. The fundamentals of Monte Carlo simulations are presented and are well-written prototypes supporting simulations. The closing Chapter, 11, is related to real-time applications in flight simulators and hardware-in-the-loop facilities. The three appendices show some matrix calculus, the CADAC primer, and several trajectory simulations. Modeling and Simulation of Aerospace Vehicle Dynamics is very well written, and it is strongly recommended to students of aerospace engineering and to practitioners in industry. The author has applied his techniques widely in research and development to rockets, missiles, aircraft, and spacecraft. His experience is documented in the book. The examples and problems are helpful and clear.

A new nutation model of a non-rigid earth with ocean and atmosphere

SUMMARY By integrating the truncated complex scalar gravitational motion equations for an anelastic, rotating, slightly elliptical earth, the complex frequency-dependent earth transfer functions are computed directly. Unlike the conventional method, the contributions of both oceanic load and current to all nutation periods, as well as the atmospheric contributions to prograde annual, retrograde annual and retrograde semi-annual nutation, are included in the integration via outer surface boundary conditions, all of which are expanded to second order in ellipticity. A modified ellipticity profile of second-order accuracy for the non-hydrostatic earth is obtained from Clairaut’s equation and the PREM earth model by adjusting both the ellipticity of the core‐mantle boundary and the global dynamical ellipticity to modern observations. The effects of different earth models, anelastic models and ocean models are computed and compared. Finally, a complete new nutation series of 343 periods, including in-phase and out-of-phase parts of longitude and obliquity terms, for a more realistic earth is obtained and compared with other available nutation series and observations.

The derivation of a scalar boundary condition for the motion of the elastic, slightly elliptical earth

The scalar equations of infinitesimal elastic-gravitational motion for a rotating, slightly elliptical Earth are always used in the theoretical study of the Earth's nutation and tides, and the determination of the integration of the equations depends, to a certain extent, on the choice of a set of boundary conditions. In this paper, a continuous quantity related to the displacement is first transformed from the elliptical reference boundary to the corresponding effective spherical domain, and is converted from vector (or tensor) form to scalar form by generalized surface spherical harmonics expansion. All the related components, including the displacement field (or the stress tensor field), are then decomposed into a poloidal and a toroidal field. After truncation, the boundary conditions are derived in scalar ordinary differential form. The derivation is given in full detail.

Towards sub-microarsecond rigid Earth nutation series in the Hamiltonian theory

AbstractThe nonrigid Earth nutation series adopted by the IAU (International Astronomical Union) are based on the works of Kinoshita (1977) and Wahr (1979). In the first one, the rigid Earth nutation series were calculated by the application of the Hamiltonian canonical equations to the rotation of the rigid and elliptical Earth. In the second one, the transfer function for the nutations of an elliptical, elastic and oceanless Earth with fluid core and a solid inner core was obtained. The nonrigid Earth nutation coefficients were derived from the convolution between Wahr’s transfer function and Kinoshita’s rigid Earth nutation series.The improvement in the accuracy of the techniques as a Very Long Baseline (VLBI), Lunar Laser Ranging (LLR) and Global Positioning System (GPS) has led in this decade to the extension of Kinoshita’s theory and more precise determination of Wahr’s transfer function. In the present paper and starting from Kinoshita’s work (1977), we present the different steps carried out, during this last decade, to obtain the sub-microarcsecond rigid Earth nutation series REN 2000 from the Hamiltonian study of the rotation of a rigid Earth (Souchay et al., 1999).

A New Nutation Model Of Nonrigid Earth With Ocean And Atmosphere

AbstractBy integrating the truncated complex scalar gravitational motion equations for an anelastic, rotating, slightly elliptical Earth, the complex frequency dependent Earth transfer functions are computed directly. Unlike the conventional method, the effects of both oceanic loads and tidal currents are included via outer surface boundary conditions, all of which are expanded to second order in ellipticity. A modified ellipticity profile in second order accuracy for the non-hydrostatic Earth is obtained from Clairaut’s equation and the PREM Earth model by adjusting both the ellipticity of the core-mantle boundary and the global dynamical ellipticity to modern observations. The effects of different Earth models, anelastic models, and ocean models are computed and compared. The atmospheric contributions to prograde annual, retrograde annual and retrograde semiannual nutation are also included as oceanic effects. Finally, a complete new nutation series of more than 340 periods, including in-phase and out-of-phase parts of longitude and obliquity terms, for a more realistic Earth, is obtained and compared with other available nutation series and observations.

HALCA phase transfer and statistical property of atmospheric phase fluctuations

Abstract The space orbiting radio telescope, HALCA, was placed into highly elliptical earth orbit on February 12, 1997. A master oscillator on board is phase locked to a reference signal generated by a hydrogen maser frequency standard in a ground tracking station. The frequency and the phase of the satellite oscillator are monitored in the ground tracking station. This paper provides a brief summary of the phase transfer link and gives the stability of the phase transfer link before and after the launch. Also this paper gives the discussions on the atmospheric phase fluctuations and the statistical property.

The Scalar Boundary Conditions For The Motion Of The Elastic Earth To Second Order In Ellipticity

The scalar equations of infinitesimal elastic gravitational motion for a rotating, slightly elliptical Earth are always used to study the Earth's nutation and tides theoretically, while the determination of the integration of the equations depends, to a certain extent, on the choice of a set of appropriate boundary conditions. In this paper, a continuity quantity related to the displacement is first transformed from the elliptical reference boundary to the corresponding effective spherical domain, and then converted from a vector (or tensor) form to a scalar form by generalized surface spherical harmonics expansion. All the related components, including the displacement vector field (or the stress tensor field), are then decomposed into the poloidal and toroidal field using the symmetry restrictions on the normal mode eigenfunctions. After truncation, the boundary conditions are finally derived, in a scalar ordinary differential form. The process of the derivation is second order in ellipticity and in full detail. Moreover, the other boundary conditions are also presented as second order in ellipticity at the end of this paper.

ARISE - A Proposed Future Space VLBI Mission

AbstractWe are developing a future space VLBI mission, ARISE, that would involve placing a VLBA-equivalent radio telescope in a highly elliptical Earth orbit. The telescope would operate at VLBI frequencies of 5, 22, 43, and 86 GHz, as well as being capable of single-dish operation at 60 GHz. Fringe-detection thresholds will be less than 10 mJy except at 86 GHz. Key scientific goals include dual-polarization imaging of blazars on the same physical scale as their γ-ray emission, astrophysical and astrometric studies of extragalactic water megamasers, and single-dish observations of O2.

Divergence of high-degree harmonic solutions of a rotating elliptical earth

Responses of a spherical or elliptical earth to external or surface forces are usually studied by expanding the equations of motion in spherical-harmonic functions. We found that, however, there exists a divergence problem in studying an elliptical earth by using harmonic expansion. This is true not only for a loading response, but also for all the other problems concerning an elliptical earth for high-degree solutions. The discussions here on the divergence at high degrees tell us that some problems (such as surface load and shear forces) could not be solved by using harmonic expansion, and prevent us from solving the problems by the same means again.

Effect of Rotation and Ellipticity On Earth Tides

SUMMARY There exists an important discrepancy in modelling earth tides on a rotating elliptical earth. the latitude dependences of Love number k2, and the gravimetric factor δ2, given by my model are significantly less than given in previous publications. My numerical procedure is checked by the analytical solution based on a simple homogeneous incompressible elastic model. the reliability of my numerical method, in this special case, is excellent. Adopting Dehant's definition of the gravimetric factor for a spheroidal earth, there is no significant latitude dependence of this parameter.

Inner Magnetosphere Imager mission: a new window on the plasma universe

The proposed Inner Magnetosphere Imager mission will obtain the first simultaneous images of the component regions of the inner magnetosphere and will enable scientists to relate these global images to internal and external influences, as well as local observations. We are performing at the George C. Marshall Space Flight Center a concept definition study of the proposed mission. The baseline mission calls for an instrument complement of approximately seven imagers to fly in an elliptical Earth orbit with an apogee of seven Earth radii (<i>Re</i>) and a perigee of approximately 4800 km. Several spacecraft concepts are being considered for the mission. The first concept utilizes a spinning spacecraft with a despun platform. The second concept splits the instruments onto two smaller satellites-a spinning spacecraft and a complementary three-axis stabilized spacecraft. Launch options being assessed for the spacecraft range from a Delta II, for the single- and dual-spacecraft concepts, to dual Taurus launches, for the two smaller spacecraft. An additional option, that of downsizing the mission to fit within the guidelines of the Space Physics Division's new class of solar terrestrial probes, is also being considered.

The ZMOA-1990 Nutation Series

AbstractWe present a new nutation series for the Earth (ZMOA-1990) based on (1) the rigid Earth nutation series developed by Zhu and Groten [1989], (2) the normalized response for an elastic, elliptical Earth with fluid-outer and solid-inner cores developed by Mathews et al. [1990], and (3) corrections for the effects of ocean tides and anelasticity, computed to be consistent with the Mathews et al. [1990] normalized response function. In deriving this series, only two parameters of the geophysical model for the Earth have been modified from their values computed with PREM: the dynamic ellipticities of the whole Earth, e, and of the fluid outer core, ef. The adopted values for these parameters, determined from the analysts of very long baseline interferometry (VLSI) data, are e=0.00328915 which is about 1% higher than the value obtained from PREM and 6×10−5 times larger than the IAU adopted value, and ef=0.002665 which is 4.6% higher than the PREM value. The above values were obtained from an adjustment of −0.3 ʺ/cent to the IAU-1976 luni-solar precession constant for e, and from the amplitude of the retrograde annual nutation for ef. The ZMOA-1990 nutation series agrees with estimates of the in-phase and the out-of-phase nutation amplitudes obtained from VLBI data to within 0.5 mas for the terms with 18.6 year period, and to better than 0.1 mas for terms at all other periods except for the out-of-phase terms with annual period (differences 0.39 mas, retrograde, and 0.13 mas, prograde), and for the in-phase term with prograde 13.66 day period (difference −0.25 mas).

The Effect of Mantle Inelasticity On Tidal Gravity: A Comparison Between the Spherical and the Elliptical Earth Model

The effects of mantle inelasticity on the Earth gravity response to the luni-solar attraction has been computed for the two most commonly used Earth models: a spherical non-rotating Earth and an elliptical uniformly rotating Earth. the corresponding results are compared in this paper. We conclude that the effects of mantle inelasticity on the amplitude of the gravimetric factor are less than 0.1 per cent for the non-zonal tides and that the effect on the tidal gravity phase is of the order of -0.005° for frequencies between the semi-diurnal and the monthly tides. These results are obtained by directly integrating the equations of motion within the Earth interior. In the spherical case they are independently verified by introducing a new method that makes use of the causality principle. Whereas the effect of mantle inelasticity on tidal gravity appears to be too small to be measured, its effect on the Chandler wobble period is significant and results in a lengthening of about 10 days.

Tidal parameters for an inelastic Earth

The gravimetric factors for an elliptical uniformly rotating Earth are computed according to a new definition with respect to Wahr's results and considering the mantle inelasticity. The Love numbers h and k are defined in the elliptical Earth case to correspond to the Love spherical definition. They become latitude dependent. The tidal parameters discussed in this paper are computed for an inelastic Earth's mantle. The numerical results show an increase of ∼ 0.4% of the gravimetric factors with respect to the Wahr's values. The Love numbers h and k increase by ∼ 1.4% to 3% with respect to the elastic case.

Partial derivatives of synthetic seismograms for a laterally heterogeneous Earth model

We present a scheme for calculating the partial derivatives with respect to elastic properties of the eigenfrequencies and eigenfunctions of a laterally heterogeneous, rotating, anelastic and elliptical earth model. We use these results to obtain a formula for the partial derivatives of synthetic seismograms (or spectra) with respect to the elastic properties. These partial derivatives, which are based on the variational solution of the forward problem, make it possible to perform iterative linearized waveform inversion for the earth's laterally heterogeneous structure.

Using delta differential one-way range to determine highly elliptical earth orbits

Acquisition strategies for delta differential one-way range measurements are designed for determining highly elliptical Earth orbits. The mission orbit for the U.S. Charged Composition Explorer spacecraft of the international Active Magnetospheric Particle Tracer Explorers serves to illustrate these acquisition techniques. The study assumes intercontinental baselines formed by Deep Space Network antennas at Goldstone, Canberra, and Madrid. The best performing strategy employs data spanning baseline view periods covering both inbound and outbound legs near the same orbit periapse. Rapidly changing viewing geometry yields both angular position and velocity information but frequently imposes the need for a different reference quasar for each observation. Although limited by an S-band transponder and a small bandwidth, the addition of delta differential one-way range to coherent Doppler and range improves apogee position accuracy by more than an order of magnitude. The measurement noise, highly dependent on acquisition geometry, varies several orders of magnitude across the baseline view periods primarily due to parallax. Careful selection of observation times minimizes the geometric influence on data noise. Additional measurement accuracy improvements, possible with dual frequency calibration, increased spanned bandwidth, and water vapor radiometry, are presented for comparison. Quasar position uncertainties then dominate the measurement noise.