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Detailed NORAD data have revealed updated orbital elements for the Ariane third-stage rocket body that underwent breakup on November 13, 1986, as well as for the Cosmos 1680 rocket body. Applying the maximum expected error due to the extrapolation of orbital elements to the date of the possible collision between the two bodies shows the smallest possible distance between bodies to have been 380 km, thereby precluding collision.
The DSAS Illumination and Moon Conflict programs are described which during an orbit when the DSAS (Digital Solar Aspect Sensor) will record the direct rays of the sun, and the periods of time when the horizon scanners will come in conflict with the moon. The DSAS Illumination Program makes use of an orbit tape (or epoch time and orbital elements) in addition to an ephemeris tape containing positions of the sun and moon. The Moon Conflict Program makes use of the same ephemeris tape with sun and moon positions, but uses only epoch time and orbital elements for the satellite positions. These programs were designed for the TIROS or ITOS series spacecraft but may be utilized by any spacecraft with similar sensors.
In the special perturbation method, Cowell and variation-of-parameters formulations of the motion equations are implemented and numerically integrated. Variations in the orbital elements due to drag are computed using the 1970 Jacchia atmospheric density model, which includes the effects of semiannual variations, diurnal bulge, solar activity, and geomagnetic activity. In the general perturbation method, two-variable asymptotic series and automated manipulation capabilities are used to obtain analytical solutions to the variation-of-parameters equations. Solutions are obtained considering the effect of oblateness only and the combined effects of oblateness and drag. These solutions are then numerically evaluated by means of a FORTRAN program in which an updating scheme is used to maintain accurate epoch values of the elements. The atmospheric density function is approximated by a Fourier series in true anomaly, and the 1970 Jacchia model is used to periodically update the Fourier coefficients. The accuracy of both methods is demonstrated by comparing computed orbital elements to actual elements over time spans of up to 8 days for the special perturbation method and up to 356 days for the general perturbation method.
TerraSAR-X (TSX) and TanDEM-X (TDX) are two advanced synthetic aperture radar (SAR) satellites flying in formation. SAR interferometry allows a high resolution imaging of the Earth by processing SAR images obtained from two slightly different orbits. TSX operates as a repeat-pass interferometer in the first phase of its lifetime and will be supplemented after two years by TDX in order to produce digital elevation models (DEM) with unprecedented accuracy. Such a flying formation makes indeed possible a simultaneous interferometric data acquisition characterized by highly flexible baselines with range of variations between a few hundreds meters and several kilometers [1]. TSX has been successfully launched on the 15th of June, 2007. TDX is expected to be launched on the 31st of May, 2009. A safe and robust maintenance of the formation is based on the concept of relative eccentricity/inclination (e/i) vector separation whose efficiency has already been demonstrated during the Gravity Recovery and Climate Experiment (GRACE) [2]. Here, the satellite relative motion is parameterized by mean of relative orbit elements and the key idea is to align the relative eccentricity and inclination vectors to minimize the hazard of a collision. Previous studies have already shown the pertinence of this concept and have described the way of controlling the formation using an impulsive deterministic control law [3]. Despite the completely different relative orbit control requirements, the same approach can be applied to the TSX/TDX formation. The task of TDX is to maintain the close formation configuration by actively controlling its relative motion with respect to TSX, the leader of the formation. TDX must replicate the absolute orbit keeping maneuvers executed by TSX and also compensate the natural deviation of the relative e/i vectors. In fact the relative orbital elements of the formation tend to drift because of the secular non-keplerian perturbations acting on both satellites. The goal of the ground segment is thus to regularly correct this configuration by performing small orbit correction maneuvers on TDX. The ground station contacts are limited due to the geographic position of the station and the costs for contact time. Only with a polar ground station a contact visibility is possible every orbit for LEO satellites. TSX and TDX use only the Weilheim ground station (in the southern part of Germany) during routine operations. This station allows two scheduled contact per day for the nominal orbit configuration, meaning that the satellite conditions can be checked with an interval of 12 hours. While this limitation is usually not critical for single satellite operations, the visibility constraints drive the achievable orbit control accuracy for a LEO formation if a ground based approach is chosen. Along-track position uncertainties and maneuver execution errors affect the relative motion and can be compensated only after a ground station contact.
Trajectory data and orbital elements for 12 double-station meteors photographed by the NASA-NMSU Meteor Observatory during the winter of 1975-1976 are presented. Three of these are members of the Quadrantid stream. Photoelectric timing of meteors, uncertainties in the derived orbital elements, and monitoring of the night sky with a two-channel photometer are discussed.
We have systematically studied the evolution of 4- and 9-micrometer-diameter dust particles from three prominent asteroid families, Eos, Koronis, and Themis, until they reach the Earth. We have found that they are quite distinguishable in their (I cos omega, I sin omega) phase space. The different distributions in the inclination space have several important observational consequences, which can be used to identify the origins of these dust particles. We convert the orbital elements of these Earth-crossing dust grains to distinguish them by using any one single velocity component. We conclude that in order to identify the origins of the Earth-crossing dust grains we need all three velocity components to get their orbital elements and find their positions in the inclination space.
Observations and results of orbit determination of the first known Mars Trojan asteroid (5261) Eureka are presented. We have numerically calculated the evolution of the orbital elements, and have analyzed the behavior of the motion during the next 2 Myr. Strong perturbations by planets other than Mars seem to stabilize the eccentricity of the asteroid by stirring the high order resonances present in the elliptic restricted problem. As a result, the orbit appears stable at least on megayear timescales. The difference of the mean longitudes of Mars and Eureka and the semimajor axis of the asteroid form a pair of variables that essentially behave in an adiabatic manner, while the evolution of the other orbital elements is largely determined by the pertubations due to other planets.
Observations obtained by the Infrared Astronomical Satellite (IRAS) during its ten month mission in 1983 were originally processed by the Asteroid Data Analysis System (ADAS) to search for 3453 asteroids with known orbital elements as of September 1985. A total of 1811 had one or more observations of sufficient reliability to be accepted. These results were released in October 1986. Recently IRAS data were reprocessed to increase both the number of recognized asteroid observations and their reliability. As input 7311 asteroids were used with known orbital elements as of December 1990. This processor is referred to as the IRAS Minor Planet Survey (IMPS). As of April 1991 approximately 3000 asteroids had been identified with one or more acceptable observations. These results were used to derive the total number of asteroids with diameters greater than 1 km. In addition to being an interesting piece of information in itself these size-frequency distributions produce bias-correction factors which, for example, will be used in investigations of the physical properties of asteroid dynamical families and to estimate the distribution of the taxonomic classes as a function of heliocentric distance.
We consider a charged dust grain whose orbital motion is dominated by a planet's point-source gravity, but perturbed by higher-order terms in the planet's gravity field as well as by the Lorentz force arising from an asymmetric planetary magnetic field. Perturbations to Keplerian orbits due to a nonspherical gravity field are expressed in the traditional way: in terms of a disturbing function which can be expanded in a series of spherical harmonics (W. M. Kaula, 1966). In order to calculate the electromagnetic perturbation, we first write the Lorentz force in terms of the orbital elements and then substitute it into Gauss' perturbation equations. We use our result to derive strengths of Lorentz resonances and elucidate their properties. In particular, we compare Lorentz resonances to two types of gravitational resonances: those arising from periodic tugs of a satellite and those due to the attraction of an arbitrarily shaped planet. We find that Lorentz resonances share numerous properties with their gravitational counterparts and show, using simple physical arguments, that several of these patterns are fundamental, applying not only to our expansions, but to all quantities expressed in terms of orbital elements. Some of these patterns have been previously called 'd'Alembert rules' for satellite resonances. Other similarities arise because, to first-order in the perturbing force, the three problems share an integral of the motion. Yet there are also differences; for example, first-order inclination resonances exist for perturbations arising from planetary gravity and from the Lorentz force, but not for those due to an orbiting satellite. Finally, we provide a heuristic treatment of a particle's orbital evolution under the influence of drag and resonant forces. Particles brought into mean-motion resonances experience either trapping or resonant 'jumps,' depending on the direction from which the resonance is approached. We show that this behavior does not depend on the details of the perturbing force but rather is fundamental to all mean-motion resonances.
Formation flying is an enabling technology for many future space missions. This paper presents extensions to the equations of relative motion expressed in Keplerian orbital elements, including new initialization techniques for general formation configurations. A new linear time-varying form of the equations of relative motion is developed from Gauss Variational Equations and used in a model predictive controller. The linearizing assumptions for these equations are shown to be consistent with typical formation flying scenarios. Several linear, convex initialization techniques are presented, as well as a general, decentralized method for coordinating a tetrahedral formation using differential orbital elements. Control methods are validated using a commercial numerical propagator.
Over the past decade, the dedicated search for kilometer-size near-Earth objects (NEOs), potentially hazardous objects (PHOs), and potential Earth impactors has led to a boost in the rate of discoveries of these objects. The catalog of known NEOs is the fundamental ingredient used to develop a model for the NEOs population, either by assessing and correcting for the observational bias (Jedicke et al., 2002), or by evaluating the migration rates from the NEOs source regions (Bottke et al., 2002). The modeled NEOs population is a necessary tool used to track the progress in the search of large NEOs (Jedicke et al., 2003) and to try to predict the distribution of the ones still undiscovered, as well as to study the sky distribution of potential Earth impactors (Chesley & Spahr, 2004). We present a method to model the NEOs population in all six orbital elements, on a finely grained grid, allowing us the design and test of targeted and optimized search strategies. This method relies on the observational data routinely reported to the Minor Planet Center (MPC) by the Catalina Sky Survey (CSS) and by other active NEO surveys over the past decade, to determine on a nightly basis the efficiency in detecting moving objects as a function of observable quantities including apparent magnitude, rate of motion, airmass, and galactic latitude. The cumulative detection probability is then be computed for objects within a small range in orbital elements and absolute magnitude, and the comparison with the number of know NEOs within the same range allows us to model the population. When propagated to the present epoch and projected on the sky plane, this provides the distribution of the missing large NEOs, PHOs, and potential impactors.
The GNC software onboard ISS utilizes TORS command loads, and a simplistic model of TORS orbital motion to generate onboard TORS state vectors. Each TORS command load contains five "invariant" orbital elements which serve as inputs to the onboard propagation algorithm. These elements include semi-major axis, inclination, time of last ascending node crossing, right ascension of ascending node, and mean motion. Running parallel to the onboard software is the TORS Command Builder Tool application, located in the JSC Mission Control Center. The TORS Command Builder Tool is responsible for building the TORS command loads using a ground TORS state vector, mirroring the onboard propagation algorithm, and assessing the fidelity of current TORS command loads onboard ISS. The tool works by extracting a ground state vector at a given time from a current TORS ephemeris, and then calculating the corresponding "onboard" TORS state vector at the same time using the current onboard TORS command load. The tool then performs a comparison between these two vectors and displays the relative differences in the command builder tool GUI. If the RSS position difference between these two vectors exceeds the tolerable lim its, a new command load is built using the ground state vector and uplinked to ISS. A command load's lifetime is therefore defined as the time from when a command load is built to the time the RSS position difference exceeds the tolerable limit. From the outset of TORS command load operations (STS-98), command load lifetime was limited to approximately one week due to the simplicity of both the onboard propagation algorithm, and the algorithm used by the command builder tool to generate the invariant orbital elements. It was soon desired to extend command load lifetime in order to minimize potential risk due to frequent ISS commanding. Initial studies indicated that command load lifetime was most sensitive to changes in mean motion. Finding a suitable value for mean motion was therefore the key to achieving this goal. This goal was eventually realized through development of an Excel spreadsheet tool called EMMIE (Excel Mean Motion Interactive Estimation). EMMIE utilizes ground ephemeris nodal data to perform a least-squares fit to inferred mean anomaly as a function of time, thus generating an initial estimate for mean motion. This mean motion in turn drives a plot of estimated downtrack position difference versus time. The user can then manually iterate the mean motion, and determine an optimal value that will maximize command load lifetime. Once this optimal value is determined, the mean motion initially calculated by the command builder tool is overwritten with the new optimal value, and the command load is built for uplink to ISS. EMMIE also provides the capability for command load lifetime to be tracked through multiple TORS ephemeris updates. Using EMMIE, TORS command load lifetimes of approximately 30 days have been achieved.
By intrack orbital error is meant a constant time adjustment that is applied to a set of ephemeris data which is otherwise correct. The ephemeris data may be in the form of an orbit tape or in the form of orbital elements with an associated orbit generator. The time adjustment is simply added to the time before the ephemeris routine is accessed. It is implicit here that the time adjustment is a constant throughout the pass of data that are considered, where the pass of data is typically a fraction of one orbit.
The atmospheric water cycle at the present epoch involves summertime sublimation of water from the north polar cap, transport of water through the atmosphere, and condensation on one or both winter CO2 caps. Exchange with the regolith is important seasonally, but the water content of the atmosphere appears to be controlled by the polar caps. The net annual transport through the atmosphere, integrated over long timescales, must be the driving force behind the long-term evolution of the polar caps; clearly, this feeds back into the evolution of the layered terrain. We have investigated the behavior of the seasonal water cycle and the net integrated behavior at the pole for the last 10 exp 7 years. Our model of the water cycle includes the solar input, CO2 condensation and sublimation, and summertime water sublimation through the seasonal cycles, and incorporates the long-term variations in the orbital elements describing the Martian orbit.
Canonical transformations are developed between the Cartesian coordinates, equinoctial elements, trajectory variables, and orbital elements for coplanar space trajectory optimization problems. The canonical transformations permit the state and adjoint or their solution, transversality conditions, the optimal control, and integrals of the motion, to be transformed between any of the common sets of coordinates for planar space trajectory optimization problems. Variations on the canonical transformations shown are straightforward to develop given the group properties of the canonical transformations.
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Decay rates and orbital elements for satellite launching
Orbital elements necessary for minimum lifetime of satellite with highly eccentric orbit are graphically specified by construction of launch window map