The remainder of this dissertation is outlined as follows. Chapter 2 presents the development of a high-fidelity geosynchronous dynamic model and numerical simulation results. A general discussion of orbit estimation approaches is given in Chapter 3, which also presents the development of a GEO element batch processor and the results of experimental and simulated orbit estimation of a single geosynchronous object. Chapter 4 presents details on the WAAS reference ephemerides and the empirical characterization of the ground-based orbit estimation accuracy. Chapter 5 develops the relative GEO element batch estimator, and demonstrates that the relative GEO elements are observable given space-based angles-only measurements and a spherical coordinate linearized Keplerian model of relative motion. This concept is expanded in Chapter 6, in which the model described in Chapter 5 is used to develop the analytic covariance models. Chapter 6 also presents the results of a Monte Carlo analysis which validate the analytic covariance profiles. Finally, Chapter 7 summarizes the dissertation research and outlines future work as an extension of this dissertation research.
Geosynchronous Dynamic Model
2.1 Introduction
The evolution of natural satellites has been studied extensively for millennia. The earliest-known study of orbital motion was performed in ancient Babylonia, as the Chaldeans developed the Saros cycle to predict solar eclipses throughout the last few centuries B.C.[29] Fundamental understanding of the kinematics of orbital motion was introduced through the combination of Tycho Brahe’s meticulous celestial observations (16th century) and Johann Kepler’s theoretical developments (early 17th century).[12] The final piece which brought together the basic theory of planetary motion was Isaac Newton’s dynamical laws (late 17th century).[102] (Interestingly, Newton’s Ph.D. thesis made no mention of his theory of gravitational forces.) Progressing forward to October 4, 1957, a new era of space exploration was born when the Soviet Union successfully launched the first man-man satellite into Earth orbit and started the space age as we know it.[3]
The first geosynchronous satellite was launched in 1963 [50], and thus began dedicated studies of the orbital motion of geosynchronous objects, whether man-made or natural.
One of the pioneering researchers who developed theory relevant to geosynchronous motion was R. R. Allan [8], whose work included the study of how the Earth’s gravitational field affects objects that are in resonance with the Earth’s rotation rate. Building on the initial gravitational work by Kaula [41], Allan developed a disturbing function to model the long-period pendulum effect on a satellite’s longitude. This disturbing function essentially models the libration about the gravity wells located at 75.1 degrees East longitude and 105.3 degrees West longitude.[90]
Geosynchronous orbit perturbation modeling was expanded by Kamel [38; 39; 40], who used a simple Hamiltonian pendulum system to approximate a low degree and order Earth gravitational field with luni-solar gravitational effects. The disturbing functions were developed for the mean equinoctial elements[38], which were then converted into osculating equinoctial elements[40]. The model was an approximation due to the limited Earth, Sun and Moon gravitational coefficients included, but provided a simple analytic model that is useful for quick consideration of two of the primary perturbations on a geosynchronous satellite. Earlier work by Kamel included the study of the effect of solar radiation pressure of the mean eccentricity of a geosynchronous orbit.[39] His work developed an analytic model of the mean eccentricity evolution and strategies for eccentricity control. A major finding of Kamel is that under the influence of solar radiation pressure, the mean eccentricity vector evolution is dependent on the spacecraft shape and size.
Throughout the past two decades, the interest in geosynchronous dynamic modeling has primarily revolved about the operational hazards introduced by debris objects in or passing through the geosynchronous orbit. Friesen, et. al. [27] simulated the motion of large and satellites in storage orbits above and below the nominal geosynchronous altitude, small debris objects initially in the circular geosynchronous orbit and small debris originating in the geosynchronous transfer orbit.
The simulations were performed over a period of 100 years. The results showed that the large satellites did not exhibit large altitude changes but did show inclination oscillation due to the luni-solar perturbations. The smaller particles, however, were primarily driven by the luni-solar radiation pressure and exhibit both eccentricity and inclination oscillations.
Most recently the long-term evolution of high area-to-mass debris objects has been studied extensively.[53; 99; 100; 101]. Valk, et. al. [100] have developed a semi-analytical Hamiltonian dynamic modeling theory which averages the geosynchronous motion over one day, and has been proven to be useful for long-term evolution studies on the order of several decades. The model included perturbations due to the J2 gravity term, luni-solar gravity, and solar radiation pressure.
The model was used to study the perturbed motion of high area-to-mass debris objects, and in particular, models coupling between the eccentricity and inclination evolution due to solar radiation
pressure. Follow-on work investigated the long-term evolution of the orbit altitude[99] and the equilibria, stability and fundamental frequencies of the resonant motion[101]. In addition to the Earth resonance, Valk, et. al. [53] introduced a secondary resonance between the object’s position and the longitude of the Sun. The impact of the secondary resonances is directly proportional to the object’s area-to-mass ratio.
In contrast to the long-period models, this research focuses on modeling the osculating el-ement short-period dynamics (on the order of several days). This chapter discusses why neither a Cartesian position and velocity nor classical orbit element representation is ideal for geosyn-chronous representation and introduces coordinates that are designed to exploit the peculiarities of the geosynchronous orbit regime. The synchronous elements, defined by Soop [90], are a specialized set defined specifically for low inclination, low eccentricity, 24-hour period orbits. The development of a geosynchronous dynamic model presented in this chapter investigates perturbation modeling with the synchronous elements and a hybrid set of synchronous and equinoctial elements (known as the GEO elements) as compared to the Keplerian and Cartesian coordinates. A nonlinear syn-chronous element dynamic model is developed for arbitrary perturbations, and the validity of small inclination and eccentricity approximations is explored. A nonlinear GEO element dynamic model is also developed and assessed. Additionally, an overview of the dominant orbital perturbations acting on geosynchronous objects is presented.