A scalable second-order analytical orbit propagator program (AOPP) is being carried out. This AOP... more A scalable second-order analytical orbit propagator program (AOPP) is being carried out. This AOPP combines modern and classical perturbation methods in function of orbit types or the requirements needed for a space mission, such as catalog maintenance operations, long period evolution, and so on. As a first step on the validation and verification of part of our AOPP, we only consider perturbation produced by zonal harmonic coefficients in the Earths gravity potential, so that it is possible to analyze the behavior of the mathematical expression involved in the corresponding analytical theory in depth and determine its limits.
Laboratory for its support of this effort. Special thanks go to Dr. Ronald J. Proulx for his assi... more Laboratory for its support of this effort. Special thanks go to Dr. Ronald J. Proulx for his assistance during the development of the plotter utility and in the interpretation of the GTDS runs using the 8 x 8 resonant harmonic field. Mr. Wayne McClain made several valuable suggestions that were helpful in interpreting the semimajor axis comparisons. Mr. Leo Early was a saving source of information on the GTDS subroutines, especially those concerning the ORBI files. It was also through his effort that the capability of the pre-processor for NORAD observations was expanded to include optical data. I'm grateful to USAF Lt. Col. Robert Herklotz, a fellow student at MIT, for his helpful comments and support. I am especially grateful to Dr. Paul J. Cefola, also of Charles Stark Draper Laboratory, who served as my thesis advisor. His patience, helpfulness, and availability were extraordinary. I'd like to express appreciation to Drs. R. Sridharan and E.M. Gaposchkin of Lincoln Laboratory for the helpful comments provided during their review of the results of this work.
Attention is given to the problem of numerical verification of orbital stability over long time s... more Attention is given to the problem of numerical verification of orbital stability over long time spans being hampered by intensive computation on fixed word length machines leading to unacceptable error growth. A third body potential in non-singular equinoctial elements, analytically double averaged with respect to both the mean longitude of the third body has been developed. In order to allow
A first-order theory is developed for the rapid and accurate calculation of secular and long-peri... more A first-order theory is developed for the rapid and accurate calculation of secular and long-period changes in the elements of a high-altitude earth orbit due to the action of the sun and moon. The third body disturbing potential is first derived in nonsingular orbital ...
Not long afterward, this model of the upper atmosphere became an integral part of the motion mode... more Not long afterward, this model of the upper atmosphere became an integral part of the motion model for ballistic support of the international Apollo-Soyuz experiment. In 1977, the government standard USSR GOST 22721-77 "Upper Atmosphere Model for Ballistic Calculations" was drafted and approved. The standard established a method of calculating atmospheric density for altitudes ranging from 120 to 600 km and for varying levels of solar activity. The standard also included a subroutine for calculating the density and temperature of Earth's atmosphere in the FORTRAN language. By 1984, through the growing volume of data on drag of AES of the Kosmos family the structure and parameters of the 1977 model were refined, and on the basis of this a new standard was established: GOST 25645.115-84, "Upper Earth Atmosphere Density Model for Ballistic Support of AES Flight." The new standard encompassed an additional range of altitudes (600- 1500 km); variations correlated ...
The Naval Space Command PPT2 model of satellite motion has been used for 30 years to maintain a c... more The Naval Space Command PPT2 model of satellite motion has been used for 30 years to maintain a catalog of Earth satellites. The PPT2 algorithm is based on Brouwer's 1959 artificial satellite theory and includes the Lyddane modification for small eccentricity and inclination and a separate modification for the critical inclination. In this paper, the authors describe a modified version of PPT2 which includes a recursive analytical model for the portion of the tesseral harmonic perturbation that depends on just the Greenwich hour angle. These tesseral m-daily terms are the major source of unmodeled periodic motion in PPT2 for many LEO satellite orbits. The recursive tesseral model is drawn from the Draper Semianalytical Satellite Theory (DSST) and provides the short-period variations in terms of the equinoctial elements [tan(i/2) convention]. Modifications to the Lyddane expressions which allow the double-primed elements to be converted to single-primed and osculating (including the J2 short periodics) equinoctial elements are introduced. The software intricacies of accessing the DSST m-daily model from the R&D GTDS version of PPT2 are addressed. Numerical comparisons of the PPT2-MDAILY algorithm with the DSST suggest that the additional errors of commission are negligible. Numerical tests with DSST as a reference are also employed to demonstrate the errors associated other unmodeled perturbations (zonal short periodics other than J2, tesseral linear combination short periodics, unmodeled zonal terms in the secular and long periodic motion, etc.). Numerical test results with Cowell as a reference are consistent with the DSST-based testing. Very high precision, externally generated reference orbits for TOPEX and TAOS are used to demonstrate the accuracy improvement associated with PPT2-MDAILY.
R 1 describes the development of a singleaveraging theory and its application to the prediction o... more R 1 describes the development of a singleaveraging theory and its application to the prediction of a near-synchronous orbit. Single averaging usually denotes a formulation in which the high-frequency perturbations associated with the satellite period and the Earth's rotational period are removed from the equations of motion. Thus, step sizes on the order of one day are allowed in the integration of the averaged equations of motion. The purpose of this Comment is to connect the results in Ref. 1 with results that have already appeared in the literature. In many cases, the author of Ref. 1 gives a limited result when a more general result is available; these instances are noted. We also discuss the construction of initial conditions for a single-averaged orbit prediction. Finally, some aspects of the very long-term motion of desynchronized orbits (such as GEOS-2) are discussed. Fundamental to the development of this single-averaged theory are the differential equations for the motion of the equinoctial elements due to a disturbing potential. These are Eqs. (3) in Ref. 1. These same equations were given much earlier in Refs. 2 and 3. The disturbing potential due to third bodies (moon and sun) is developed in Ref. 1 with the aid of a Poisson series symbolic algebra program. The potentials include "up to second order eccentricity terms." The author of Ref. 1 notes that "lengthy computer-generated secular terms can be rearranged in extremely compact forms" by introducing the C and S auxiliary parameters [Eq. (17) in Ref. 1 ]. The averaged potentials are obtained under the assumption that the third body positions are held constant during the averaging operation. Similar potentials were given much earlier in Refs. 3 and 5. In particular, Ref. 5 gave the following general form for the third body potential in terms of equinoctial elements:
A second-order closed-form semi-analytical solution of the main problem of the artificial satelli... more A second-order closed-form semi-analytical solution of the main problem of the artificial satellite theory ($$J_2$$ J 2 contribution) consistent with the Draper Semi-analytic Satellite Theory (DSST) is presented. This paper aims to improve the computational speed of the numerical-based approach, which is only available in the GTDS-DSST version. The short-period terms are removed by means of an extension of the Lie-Deprit method using Delaunay variables. The averaged equations of motion are given explicitly and transformed to the non-singular equinoctial elements. Finally, the second-order terms in the equations of motion are included in the C/C++ version of the DSST orbit propagator.
The study presents an analysis of the orbit determination process for maintenance of the space ob... more The study presents an analysis of the orbit determination process for maintenance of the space object catalogue. After quantifying the uncertainties in optical and radar measurements, the second part of the study presents the error within the mean element orbit determination procedure. A batch least square orbit determination method which makes use of the Draper semi-analytical satellite theory and its partial derivatives is tested for its correctness under catalogue maintenance conditions. Different attributes, which could affect the performance of such an orbit determination method, are identified and examined with real and simulated observations. In conclusion, the factors which limit the practical accuracies of the orbit determination procedure are discussed.
The study presents an analysis of accuracy of observations which are contributing to the maintena... more The study presents an analysis of accuracy of observations which are contributing to the maintenance of space object catalogue. After quantifying the uncertainties in optical and radar measurements, the second part of the study presents the error within the mean element orbit determination procedure. A batch least square orbit determination method which makes use of the Draper semi-analytical satellite theory and its partial derivatives is tested for its correctness under catalogue maintenance conditions. Different attributes, which could affect the performance of such an orbit determination method, are identified and examined with real and simulated observations. In conclusion, an analysis of the factors which limit the practical accuracies of orbit determination are discussed.
Catalog maintenance for Space Situational Awareness (SSA) demands an accurate and computationally... more Catalog maintenance for Space Situational Awareness (SSA) demands an accurate and computationally lean orbit propagation and orbit determination technique to cope with the ever increasing number of observed space objects. As an alternative to established numerical and analytical methods, we investigate the accuracy and computational load of the Draper Semi-analytical Satellite Theory (DSST). The standalone version of the DSST was enhanced with additional perturbation models to improve its recovery of short periodic motion. The accuracy of DSST is, for the first time, compared to a numerical propagator with fidelity force models for a comprehensive grid of low, medium, and high altitude orbits with varying eccentricity and different inclinations. Furthermore, the run-time of both propagators is compared as a function of propagation arc, output step size and gravity field order to assess its performance for a full range of relevant use cases. For use in orbit determination, a robust performance of DSST is demonstrated even in the case of sparse observations, which is most sensitive to mismodeled short periodic perturbations. Overall, DSST is shown to exhibit adequate accuracy at favorable computational speed for the full set of orbits that need to be considered in space surveillance. Along with the inherent benefits of a semianalytical orbit representation, DSST provides an attractive alternative to the more common numerical orbit propagation techniques.
An improved accuracy model for the J2 zonal harmonic second order terms, J 2 2 , in a general sem... more An improved accuracy model for the J2 zonal harmonic second order terms, J 2 2 , in a general semi-analytical satellite theory is described. The new model is implemented as an option in the Draper Semi-analytic Satellite Theory (DSST) orbit propagator; the DSST is one of the several orbit propagators included in the Linux version of the R&D Goddard Trajectory Determination System (GTDS) program. The DSST previously modeled the first-order mean equinoctial orbital element rates and short periodic motion for the Earth's zonal and tesseral gravity harmonics. The current DSST also includes approximate models for the mean equinoctial orbit element rates and the short-periodic motion due to the J 2 2 zonal harmonic terms. However, these approximate J 2 2 models are truncated on the eccentricity. Building on the work of McClain, Zeis, Slutsky, and Fischer, we develop a method to calculate the J 2 2 mean element rates using Gauss-Kronrod numerical quadrature. This quadrature-based method has both analytical and numerical components. The open source symbolic algebra system, Maxima, operating on a Linux platform, is used to construct the analytical portion of the model. The accuracy of both the old and new J 2 2 models is demonstrated over a range of eccentricities and inclinations.
With an increasing dependence on space systems in all areas of society, the need for accurate and... more With an increasing dependence on space systems in all areas of society, the need for accurate and robust Space Situational Awareness (SSA) is paramount. Despite this, software tools have been slow to progress; many systems remain in legacy languages, based on deprecated design principles. The Draper Semianalytic Satellite Theory (DSST) is one such system, though especially suited to the problem of SSA due to its high computational efficiency. Originally implemented in Fortran 77 in the proprietary R&D Goddard Trajectory Determination System (GTDS), DSST has remained inaccessible to most researchers. Recent work has been undertaken to reimplement DSST in the open source ORbit Extrapolation KIT (OREKIT), a modern and operationally tested astrodynamics library. Tests to verify OREKIT DSST’s accuracy and stability will be presented, including direct comparisons with the GTDS implementation. It is hoped that this, and other additions to OREKIT, will provide a modern, verified, and freely...
Verification of the java Orekit implementation of the Draper Semi-analytical Satellite Theory (DS... more Verification of the java Orekit implementation of the Draper Semi-analytical Satellite Theory (DSST) is discussed. The Orekit library for space flight dynamics has been published under the open-source Apache license V2. The DSST is unique among analytical and semi-analytical satellite theories due to the scope of the included force models. However, the DSST has not been readily accessible to the wider Astrodynamics research community. Implementation of the DSST in the Orekit library is a comprehensive task because it involves the migration of the DSST to the object-oriented java language and to a different functional decomposition strategy. The resolution of the code and documentation anomalies discovered during the verification process is the important product of this project.
A scalable second-order analytical orbit propagator program (AOPP) is being carried out. This AOP... more A scalable second-order analytical orbit propagator program (AOPP) is being carried out. This AOPP combines modern and classical perturbation methods in function of orbit types or the requirements needed for a space mission, such as catalog maintenance operations, long period evolution, and so on. As a first step on the validation and verification of part of our AOPP, we only consider perturbation produced by zonal harmonic coefficients in the Earths gravity potential, so that it is possible to analyze the behavior of the mathematical expression involved in the corresponding analytical theory in depth and determine its limits.
Laboratory for its support of this effort. Special thanks go to Dr. Ronald J. Proulx for his assi... more Laboratory for its support of this effort. Special thanks go to Dr. Ronald J. Proulx for his assistance during the development of the plotter utility and in the interpretation of the GTDS runs using the 8 x 8 resonant harmonic field. Mr. Wayne McClain made several valuable suggestions that were helpful in interpreting the semimajor axis comparisons. Mr. Leo Early was a saving source of information on the GTDS subroutines, especially those concerning the ORBI files. It was also through his effort that the capability of the pre-processor for NORAD observations was expanded to include optical data. I'm grateful to USAF Lt. Col. Robert Herklotz, a fellow student at MIT, for his helpful comments and support. I am especially grateful to Dr. Paul J. Cefola, also of Charles Stark Draper Laboratory, who served as my thesis advisor. His patience, helpfulness, and availability were extraordinary. I'd like to express appreciation to Drs. R. Sridharan and E.M. Gaposchkin of Lincoln Laboratory for the helpful comments provided during their review of the results of this work.
Attention is given to the problem of numerical verification of orbital stability over long time s... more Attention is given to the problem of numerical verification of orbital stability over long time spans being hampered by intensive computation on fixed word length machines leading to unacceptable error growth. A third body potential in non-singular equinoctial elements, analytically double averaged with respect to both the mean longitude of the third body has been developed. In order to allow
A first-order theory is developed for the rapid and accurate calculation of secular and long-peri... more A first-order theory is developed for the rapid and accurate calculation of secular and long-period changes in the elements of a high-altitude earth orbit due to the action of the sun and moon. The third body disturbing potential is first derived in nonsingular orbital ...
Not long afterward, this model of the upper atmosphere became an integral part of the motion mode... more Not long afterward, this model of the upper atmosphere became an integral part of the motion model for ballistic support of the international Apollo-Soyuz experiment. In 1977, the government standard USSR GOST 22721-77 "Upper Atmosphere Model for Ballistic Calculations" was drafted and approved. The standard established a method of calculating atmospheric density for altitudes ranging from 120 to 600 km and for varying levels of solar activity. The standard also included a subroutine for calculating the density and temperature of Earth's atmosphere in the FORTRAN language. By 1984, through the growing volume of data on drag of AES of the Kosmos family the structure and parameters of the 1977 model were refined, and on the basis of this a new standard was established: GOST 25645.115-84, "Upper Earth Atmosphere Density Model for Ballistic Support of AES Flight." The new standard encompassed an additional range of altitudes (600- 1500 km); variations correlated ...
The Naval Space Command PPT2 model of satellite motion has been used for 30 years to maintain a c... more The Naval Space Command PPT2 model of satellite motion has been used for 30 years to maintain a catalog of Earth satellites. The PPT2 algorithm is based on Brouwer's 1959 artificial satellite theory and includes the Lyddane modification for small eccentricity and inclination and a separate modification for the critical inclination. In this paper, the authors describe a modified version of PPT2 which includes a recursive analytical model for the portion of the tesseral harmonic perturbation that depends on just the Greenwich hour angle. These tesseral m-daily terms are the major source of unmodeled periodic motion in PPT2 for many LEO satellite orbits. The recursive tesseral model is drawn from the Draper Semianalytical Satellite Theory (DSST) and provides the short-period variations in terms of the equinoctial elements [tan(i/2) convention]. Modifications to the Lyddane expressions which allow the double-primed elements to be converted to single-primed and osculating (including the J2 short periodics) equinoctial elements are introduced. The software intricacies of accessing the DSST m-daily model from the R&D GTDS version of PPT2 are addressed. Numerical comparisons of the PPT2-MDAILY algorithm with the DSST suggest that the additional errors of commission are negligible. Numerical tests with DSST as a reference are also employed to demonstrate the errors associated other unmodeled perturbations (zonal short periodics other than J2, tesseral linear combination short periodics, unmodeled zonal terms in the secular and long periodic motion, etc.). Numerical test results with Cowell as a reference are consistent with the DSST-based testing. Very high precision, externally generated reference orbits for TOPEX and TAOS are used to demonstrate the accuracy improvement associated with PPT2-MDAILY.
R 1 describes the development of a singleaveraging theory and its application to the prediction o... more R 1 describes the development of a singleaveraging theory and its application to the prediction of a near-synchronous orbit. Single averaging usually denotes a formulation in which the high-frequency perturbations associated with the satellite period and the Earth's rotational period are removed from the equations of motion. Thus, step sizes on the order of one day are allowed in the integration of the averaged equations of motion. The purpose of this Comment is to connect the results in Ref. 1 with results that have already appeared in the literature. In many cases, the author of Ref. 1 gives a limited result when a more general result is available; these instances are noted. We also discuss the construction of initial conditions for a single-averaged orbit prediction. Finally, some aspects of the very long-term motion of desynchronized orbits (such as GEOS-2) are discussed. Fundamental to the development of this single-averaged theory are the differential equations for the motion of the equinoctial elements due to a disturbing potential. These are Eqs. (3) in Ref. 1. These same equations were given much earlier in Refs. 2 and 3. The disturbing potential due to third bodies (moon and sun) is developed in Ref. 1 with the aid of a Poisson series symbolic algebra program. The potentials include "up to second order eccentricity terms." The author of Ref. 1 notes that "lengthy computer-generated secular terms can be rearranged in extremely compact forms" by introducing the C and S auxiliary parameters [Eq. (17) in Ref. 1 ]. The averaged potentials are obtained under the assumption that the third body positions are held constant during the averaging operation. Similar potentials were given much earlier in Refs. 3 and 5. In particular, Ref. 5 gave the following general form for the third body potential in terms of equinoctial elements:
A second-order closed-form semi-analytical solution of the main problem of the artificial satelli... more A second-order closed-form semi-analytical solution of the main problem of the artificial satellite theory ($$J_2$$ J 2 contribution) consistent with the Draper Semi-analytic Satellite Theory (DSST) is presented. This paper aims to improve the computational speed of the numerical-based approach, which is only available in the GTDS-DSST version. The short-period terms are removed by means of an extension of the Lie-Deprit method using Delaunay variables. The averaged equations of motion are given explicitly and transformed to the non-singular equinoctial elements. Finally, the second-order terms in the equations of motion are included in the C/C++ version of the DSST orbit propagator.
The study presents an analysis of the orbit determination process for maintenance of the space ob... more The study presents an analysis of the orbit determination process for maintenance of the space object catalogue. After quantifying the uncertainties in optical and radar measurements, the second part of the study presents the error within the mean element orbit determination procedure. A batch least square orbit determination method which makes use of the Draper semi-analytical satellite theory and its partial derivatives is tested for its correctness under catalogue maintenance conditions. Different attributes, which could affect the performance of such an orbit determination method, are identified and examined with real and simulated observations. In conclusion, the factors which limit the practical accuracies of the orbit determination procedure are discussed.
The study presents an analysis of accuracy of observations which are contributing to the maintena... more The study presents an analysis of accuracy of observations which are contributing to the maintenance of space object catalogue. After quantifying the uncertainties in optical and radar measurements, the second part of the study presents the error within the mean element orbit determination procedure. A batch least square orbit determination method which makes use of the Draper semi-analytical satellite theory and its partial derivatives is tested for its correctness under catalogue maintenance conditions. Different attributes, which could affect the performance of such an orbit determination method, are identified and examined with real and simulated observations. In conclusion, an analysis of the factors which limit the practical accuracies of orbit determination are discussed.
Catalog maintenance for Space Situational Awareness (SSA) demands an accurate and computationally... more Catalog maintenance for Space Situational Awareness (SSA) demands an accurate and computationally lean orbit propagation and orbit determination technique to cope with the ever increasing number of observed space objects. As an alternative to established numerical and analytical methods, we investigate the accuracy and computational load of the Draper Semi-analytical Satellite Theory (DSST). The standalone version of the DSST was enhanced with additional perturbation models to improve its recovery of short periodic motion. The accuracy of DSST is, for the first time, compared to a numerical propagator with fidelity force models for a comprehensive grid of low, medium, and high altitude orbits with varying eccentricity and different inclinations. Furthermore, the run-time of both propagators is compared as a function of propagation arc, output step size and gravity field order to assess its performance for a full range of relevant use cases. For use in orbit determination, a robust performance of DSST is demonstrated even in the case of sparse observations, which is most sensitive to mismodeled short periodic perturbations. Overall, DSST is shown to exhibit adequate accuracy at favorable computational speed for the full set of orbits that need to be considered in space surveillance. Along with the inherent benefits of a semianalytical orbit representation, DSST provides an attractive alternative to the more common numerical orbit propagation techniques.
An improved accuracy model for the J2 zonal harmonic second order terms, J 2 2 , in a general sem... more An improved accuracy model for the J2 zonal harmonic second order terms, J 2 2 , in a general semi-analytical satellite theory is described. The new model is implemented as an option in the Draper Semi-analytic Satellite Theory (DSST) orbit propagator; the DSST is one of the several orbit propagators included in the Linux version of the R&D Goddard Trajectory Determination System (GTDS) program. The DSST previously modeled the first-order mean equinoctial orbital element rates and short periodic motion for the Earth's zonal and tesseral gravity harmonics. The current DSST also includes approximate models for the mean equinoctial orbit element rates and the short-periodic motion due to the J 2 2 zonal harmonic terms. However, these approximate J 2 2 models are truncated on the eccentricity. Building on the work of McClain, Zeis, Slutsky, and Fischer, we develop a method to calculate the J 2 2 mean element rates using Gauss-Kronrod numerical quadrature. This quadrature-based method has both analytical and numerical components. The open source symbolic algebra system, Maxima, operating on a Linux platform, is used to construct the analytical portion of the model. The accuracy of both the old and new J 2 2 models is demonstrated over a range of eccentricities and inclinations.
With an increasing dependence on space systems in all areas of society, the need for accurate and... more With an increasing dependence on space systems in all areas of society, the need for accurate and robust Space Situational Awareness (SSA) is paramount. Despite this, software tools have been slow to progress; many systems remain in legacy languages, based on deprecated design principles. The Draper Semianalytic Satellite Theory (DSST) is one such system, though especially suited to the problem of SSA due to its high computational efficiency. Originally implemented in Fortran 77 in the proprietary R&D Goddard Trajectory Determination System (GTDS), DSST has remained inaccessible to most researchers. Recent work has been undertaken to reimplement DSST in the open source ORbit Extrapolation KIT (OREKIT), a modern and operationally tested astrodynamics library. Tests to verify OREKIT DSST’s accuracy and stability will be presented, including direct comparisons with the GTDS implementation. It is hoped that this, and other additions to OREKIT, will provide a modern, verified, and freely...
Verification of the java Orekit implementation of the Draper Semi-analytical Satellite Theory (DS... more Verification of the java Orekit implementation of the Draper Semi-analytical Satellite Theory (DSST) is discussed. The Orekit library for space flight dynamics has been published under the open-source Apache license V2. The DSST is unique among analytical and semi-analytical satellite theories due to the scope of the included force models. However, the DSST has not been readily accessible to the wider Astrodynamics research community. Implementation of the DSST in the Orekit library is a comprehensive task because it involves the migration of the DSST to the object-oriented java language and to a different functional decomposition strategy. The resolution of the code and documentation anomalies discovered during the verification process is the important product of this project.
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Papers by Paul Cefola