Introduction: we are challenged and motivated by the increasing requirements for accuracy and reliability for satellite orbit models. It is clear to us that the increasing demands for improvements over currently available satellite orbit models are significant and must be met. We are convinced that the requirements for accuracy and reliability can only be obtained by adhering strictly to governing physical laws with special emphasize being placed on Newton's law of gravitation. We further believe that data reduction schemes of satellite ranging measurements and satellite GPS measurements must be performed using reliable algorithms. Finally, that known physical parameters and measurements that contribute to gravitational modeling must be incorporated with the highest fidelity. That is why we have chosen to fully represent the topography of the earth's surface in gravitational modeling instead of relying on a nominal spherical representation. Our approach to satellite orbit modeling differs from all the others we have seen in that we are fully committed to satisfying Newton's law of gravitation and the accurate representation of earth's surface's topography entailed by this commitment.
Adherence to Reality: the reasons we start with the correct shape for the earth is that we are committed to what we regard as the correct decision to represent the earth's gravitational field as a product integral over the volume of the earth. One of the factors in this product is the familiar inverse distance squared function of Newton's law of gravitation. The other factor is the earth's mass distribution. This representation requires us to have an accurate representation of the earth's topography. Finally, we apply rigorous error checking before releasing a satellite orbit models to verify that Newton's law of gravitation is, indeed, satisfied. This distinguishes our orbit models from the customary orbit models that are readily available and widely employed.
The Problem: in accordance with Newton's law of gravitation, the effects of the earth's gravitational field on the trajectory of a satellite is represented by a multitude of product integrals. The domain of these integrals is the volume of the earth. One factor in these product integrals is the inverse distance squared function of Newton's law of gravitation. The other factor, of course, is the earth's mass distribution. The success of orbit mechanics methodologies for artificial satellites are largely dependent on algorithms for inverting these product integrals. Such an inversion process would resolve those components of the earth's mass distribution that shape satellite orbit trajectories. Such algorithms are essential to establish accurate and reliable methodologies for determining and predicting satellite orbit trajectories.
Indications of the problem: there are multiple cases that point towards this avoidance of modeling the effects of gravitation in accordance with Newton's law of gravitation. The fact that the positions and velocities of operational satellites are constantly monitored and updated indicate that current satellite orbit models are incomplete and hence unable to track the satellites over extended time periods. Another indication is the constant corrections made to the heights of satellites measuring the topography of the ocean surfaces. Finally, the difficulties in modeling the radiation effects of the sun as satellites enter the shadow of the earth is an indication that there are constant timing errors as to when satellites enter and leave these shadow regions.
Previous solutions to these problems: Kepler and Newton successfully modeled the movements of the planets in the solar system. Their success and the continued success of subsequent astronomers depended on estimates of the relative masses of the planets and the sun. And, in fact, these estimates are improved as better measurement techniques become available.
What everyone has agreed to ignore is that attributing all of the observed motions of the planets around the sun to these relative masses avoids the challenging problem of representing these gravitational effects with product integrals over the volumes of these bodies. This simplification was started by Newton when he chose to approximate the gravitational fields of these bodies with the gravitational field of pinpoint masses located at the center of these bodies. His choice was astute and certainly has withstood the test of time.
Proposed solution: however, the problem remains with us. The problem is complex. However, in this day and age we have the advantages of powerful computer resources and ongoing developments in numerical algorithms. These algorithms include improved and more efficient methods for determining the singular value decomposition (SVD) of large-scale matrices. In addition, these algorithms are sufficiently general so that they can be conveniently applied to locally convex spaces. This combination of computer resources and modern algorithmic methods now make the solution of this problem feasible.
Effectiveness and applications of solution: the first application of these solution techniques is to establish a quantitative framework for satellite orbit models. The explicit determination of constraints on the distribution of the mass of the earth enables engineers and scientists to perform rigorous error analyses on the processes of orbit determination and orbit propagation. Furthermore, the gravitational constraints generated in the orbit fitting process of a satellite increases the overall accuracy of the gravitational modeling processes. It results than that there will be a collective improvement in satellite orbit modeling as measurements are gathered.
The same methodologies have additional applications. These methodologies can be applied to MRI and CRT images in medicine. Furthermore, the same methodologies will change antenna design and enhance signal transmission and the recovery of remote sensing data.
Gravitational models that are widely employed: spherical harmonics are widely used in satellite orbit models. For that reason, we discuss some of the problems and concerns about the use of these methodologies. The fitting processes that are employed to determine estimates for the spherical harmonic coefficients essentially are identical to fitting processes that are customarily employed to fit gravimetric data that is gathered on the surface of the earth. These methodologies have been transplanted into outer space without necessary changes or modifications.
This transplanting, of course, is widespread and commonly practiced. Our concern about this unadulterated transplanting of technology which was created for different purposes is that one always has to examine and critique whether or not the new, it is necessary for us to address some of the issues of spherical harmonic gravitational methodologies. These methodologies are the displaced methodologies that ordinarily are applied to fit gravimetric measurements gathered nearby or at the earth's surface. These methodologies are essentially two-dimensional. Of course, there are corrective mechanisms to account for variations in height due to the topography of the earth. However, these methodologies have been required to handle the three-dimensional problem of modeling the effects of the earth's gravitational field on orbiting satellites even though no changes have been included to account for the differences between the two different fitting requirements.
Innovative Technology: when one creates satellite orbit models, it is necessary to model the effects of the earth's gravitational field fully and accurately. Newton's law of gravitation tells us that the correct representation of the earth's gravitational field is a product integral over the volume of the earth. One factor in this product is the familiar inverse distance squared function of Newton's law of gravitation. The other factor is the earth's mass distribution.
Now the earth's mass distribution is the unknown in satellite orbit models. The effects of this large and significant mass distribution must be determined. We determine these effects by inverting the product integral over the earth's volume. Because of the large number of equations involved in parameterizing both the earth's mass distribution and the satellite's orbit trajectory, inverting these equations requires numerical a intensive, iteration processes. The exact nature of these processes is proprietary to SSI.
Accuracy and Reliability: however, both the accuracy and reliability are checked by innovative methodologies created by SSI that are now in the public domain. These methodologies essentially check that both Newton's law of gravitation is satisfied and that the correct shape of the earth is being utilized.