Assuming that the measurements are reliable, the discrepancies between the computed observables and

The orbit determination problem has its origin in the early efforts of solar system astronomers attempting to describe the motions of the planets and comets as they orbit the Sun . In these studies, the observations of the bodies, as observed from the surface of Earth, were fit to a path through the heavens that was completely described by six parameters. The foundations of modern estimation theory evolved from the early attempts to develop techniques to determine the six fundamental orbital parameters. Three of the parameters determine the orientation of the orbit or trajectory plane in space, and three locate the body in the orbital plane. These six parameters are uniquely related to the position and velocity of the satellite at a given epoch. hotels in downtown memphis Six appropriately selected observations will yield a solution for the trajectory. This is the classic orbit determination problem in which there is a match between the number of observations and the parameters to be determined. However, when the number of observations exceeds hotels in downtown memphis the number of parameters to be assigned, special techniques were required to allow using all observations. One solution to this problem was the method of least squares, which was proposed by Gauss in 1795 before his twentieth hotels in downtown memphis birthday. In an independent study, Legendre published a similar method in 1806 that led to considerable early debate about who originated the method. This early activity was followed by a period of intense study, culminating in the current theory for estimating dynamic parameters using observations hotels in downtown memphis corrupted by random measurement error.
During the past two decades, the requirements hotels in downtown memphis for highly accurate determinations of the orbits of near-Earth satellites have been driven by the evolution of the fields of satellite geodesy and satellite hotels in downtown memphis oceanography. The ability to use satellite altimeter measurements to obtain accurate, hotels in downtown memphis globally distributed, and temporally dense observations of a satellite height as it traverses the ocean surface has opened a new era in oceanography. The ability to use accurate range and range-rate measurements between hotels in downtown memphis an orbiting satellite and tracking systems located on Earth s surface hotels in downtown memphis has provided hotels in downtown memphis a dramatic improvement in the ability to monitor tectonic surface deformation and subsidence and to monitor small but important changes in Earth s rotation. These same measurements, along with satellite-to-satellite measurements, are providing hotels in downtown memphis unparalleled views of Earth s gravity field and the gravitational signals from temporal variations in Earth s mass distribution. The advances in each of these fields is tied to the advances hotels in downtown memphis in our ability to determine, at high accuracy, the path followed by an Earth-orbiting satellite. The methodology hotels in downtown memphis whereby this task is accomplished is referred to as precision hotels in downtown memphis orbit determination (POD).
In alternative applications, hotels in downtown memphis the recent advances in the spatial resolution of orbiting microwave and multispectral imaging systems has stimulated the requirements for accurate near-real-time orbits. These requirements along with the operational challenges of space object catalog maintenance to support collision hotels in downtown memphis avoidance in spacecraft operations has stimulated the need for accurate orbit prediction. The improvements in our knowledge of the models for the forces that influence a satellite s motion along with the dramatic improvement in computational capability has opened a new era in determining hotels in downtown memphis and predicting the orbits of near-Earth satellites.
The solution of the orbit determination problem involves four fundamental elements: (1) a set of differential equations that describes the motion of the satellite; (2) a numerical hotels in downtown memphis integration procedure to obtain a solution of the differential equations; (3) accurate observations of the satellite s motion; and (4) an appropriate estimation method that combines the results of the first three to yield an estimate of the satellite s position, velocity, and appropriate model parameters (e.g., the drag coefficient). The basic procedure starts with an initial model of the trajectory of a satellite during some time interval. This initial orbit will be incorrect due to errors in the estimate for the starting point, deficiencies in the mathematical model for the forces hotels in downtown memphis acting on the satellite, and errors in the parameters used in the model. To correct the model, independent observations of the satellite s motion must be obtained. These observations generally measure only some component of the motion, such as the distance or rate of change of the distance between the satellite and a ground-based tracking station. Measurements of the full three-dimensional position or velocity are usually not available, but as long as the observations depend on the satellite s motion, they contain information that helps to determine the orbit. The evolution of the satellite s position and velocity must be consistent with both the physics of the mathematical model and the sequence of observations, which constrains the orbit estimate to a specific solution.
The observations must also have a corresponding mathematical model to be usable in the orbit estimation problem. The observation model depends on the satellite s motion, and also on the orientation of the spacecraft hotels in downtown memphis and the motion of the observing station. The measurement model must relate hotels in downtown memphis the location of the tracking instrument to the spacecraft hotels in downtown memphis s center of mass, which may change with time as onboard fuel is consumed. At the same time, the tracking station hotels in downtown memphis is on a rotating Earth. The observing station may even be another orbiting satellite, such as a Global Positioning Satellite (GPS). Finally, the measurement model must account for various atmospheric refractive effects and other instrument effects.
Assuming that the measurements are reliable, the discrepancies between the computed observables and the real observations (called the residuals) contain measurement errors and the effects of errors in the initial conditions as well as deficiencies in the dynamic and observational models. Through a linearized least-squares solution process discussed later, the initial conditions and selected model parameters are adjusted to minimize the residuals. Considerable hotels in downtown memphis experience is required in choosing the model parameters that are best suited for adjustment. The mathematical models hotels in downtown memphis will always be imperfect in some respects, and the adjusted parameters are chosen for their ability to compensate for the deficiencies. The orbit is then recalculated on the basis of improved initial conditions and parameters, the observations are again compared with their computed counterparts, and the initial hotels in downtown memphis conditions and parameters are adjusted again. During this iterative process, unreliable observations can be identified and removed. Given a set of observations that contain sufficient information, the adjustments become smaller and smaller with each iteration, and the process is judged to have converged when a satisfactory and stable orbit solution is obtained.
The advent of the GPS and space-qualified GPS receivers have allowed continuous kinematic (i.e., purely geometric) hotels in downtown memphis positioning of satellites. However, the dynamic techniques discussed here still tend to provide the best results.
The precise trajectory of a satellite has generally been obtained by integrating Newton s dynamic equations of motion by numerical methods (1). The mathematical representation of the motion of the center of mass of a spacecraft is given by
where t is time; r and v are the position and velocity vectors of the spacecraft s center of mass whose initial values are r0 and V0 at time to; ~ is the acceleration (force per unit mass) of the spacecraft; and p represents all of the parameters that are employed in the models for the reference frame, the forces, and the observations. The arc length is the time interval from the initial point to some chosen final time. This may be several hours, a day, several days, or longer.
The forces acting on the satellite can be broadly classified as either gravitational or nongravitational. Among the gravitational forces, the two-body term (where the central body is assumed to be perfectly spherical) dominates the orbital motion hotels in downtown memphis by far. As a consequence, an orbit is well characterized by the Keplerian elements of an elliptical orbit (2,3). Figure 1 illustrates the geometric properties of the usual set of orbital elements used to describe the motion of a satellite in orbit about the Earth. The satellite height is characterized by the orbit s semimajor axis a, the variation in the radial hotels in downtown memphis distance due to the ellipticity of the orbit (the eccentricity e), and the angular distance n (the true anomaly) from the point of closest approach in the orbit (called the perigee). hotels in downtown memphis Other angular measures hotels in downtown memphis besides n are used to relate time and the motion of the satellite along the orbit, including the eccentric anomaly E and the mean anomaly M. The tilt and orientation of the orbital plane are given by the inclination i and the longitude of the ascending node O. These two angles are related to the out-of-plane components of the orbit. The argument of perigee w is the angular distance along the orbit from the equatorial plane to the perigee, which determines the orientation of the long axis of the elliptical orbit within the orbital plane. The motion of a satellite in Earth orbit is principally characterized hotels in downtown memphis by these six orbital elements; the satellite is moving along an elliptical orbit within a plane that tends to precess slowly in space.
The Earth s gravity field is, however, not perfectly spherical, and undulations hotels in downtown memphis in the gravity field, corresponding to the variations in Earth s shape and density (discussed in other articles), cause perturbations from perfectly elliptical motion (4,5). In addition, perturbations are caused by the gravitational attraction of the Sun, Moon, and planets. For highly precise applications, the effects of general relativity, principally the precession of perigee and the relativisti