The Main Problem in Satellite Theory Revisited

Abstract. Using the elimination of the parallax followed by the Delaunay normalization, we present a procedure for calculating a normal form of the main problem (J2 perturbation only) in satellite theory. This procedure is outlined in such a way that an object-oriented automatic symbolic manipulator based on a hierarchy of algebras can perform this computation. The Hamiltonian after the Delaunay normalization is presented to order six explicitly in closed form, that is, in which there is no expansion in the eccentricity. The corresponding generating function and transformation of coordinates, too lengthy to present here to the same order; the generator is given through order four

Orbit propagation with Lie transfer maps in the perturbed Kepler problem

The Lie transfer map method may be applied to orbit propagation problems in celestial mechanics. This method, described in another paper, is a perturbation method applicable to Hamiltonian systems. In this paper, it is used to calculate orbits for zonal perturbations to the Kepler (two-body) problem, in both expansion in the eccentricity and closed form. In contrast with a normal form method like that of Deprit, the Lie transformations here are used to effect a propagation of phase space in time, and not to transform one Hamiltonian into another

Student Projects for Space Navigation and Guidance

"Space Navigation and Guidance," taught every fall at the University of Maryland, is required of all space track undergraduate aerospace engineering majors. Every student is required to participate in a group project where real observations are used in the solution of a navigation problem with estimation from observations. In this paper, I discuss two such projects, an observatory project in which the students use a telescope to track a satellite and determine its orbit, and a GPS project in which they analyze GPS receiver data to determine the receiver's position

Speed and Accuracy Tests of the Variable-Step Störmer-Cowell Integrator

See also the dissertation of Matt Berry http://scholar.lib.vt.edu/theses/available/etd-04282004-071227/.The variable-step Stormer-Cowell integrator is a non-summed, double-integration multi-step integrator derived in variable-step form. The method has been implemented with a Shampine-Gordon style error control algorithm that uses an approximation of the local error at each step to choose the step size for the subsequent step. In this paper, the variable-step Stormer-Cowell method is compared to several other multi-step integrators, including the fixed-step Gauss-Jackson method, the Gauss-Jackson method with s-integration, and the variable-step single-integration Shampine- Gordon method, in both orbit propagation and orbit determination. The results show the variable-step Stormer-Cowell method is comparable with Gauss-Jackson using s-integration, except in high drag cases where the variable-step Stormer-Cowell method has an advantage in speed and accuracy

The generalized Sundman transformation for propagation of high-eccentricity elliptical orbits

See also dissertation of Matt Berry at http://scholar.lib.vt.edu/theses/available/etd-04282004-071227/A generalized Sundman transformation dt = cr^n ds for exponent n >= 1 may be used to accelerate the numerical computation of high-eccentricity orbits, by transforming time t to a new independent variable s. Once transformed, the integration in uniform steps of s effectively gives analytic step variation in t with larger time steps at apogee than at perigee, making errors at each point roughly comparable. In this paper, we develop techniques for assessing accuracy of s-integration in the presence of perturbations, and analyze the effectiveness of regularizing the transformed equations. A computational speed comparison is provided

Implementation of Gauss-Jackson Integration for Orbit Propagation

This paper focuses on the Gauss-Jackson algorithm for numerical integration, which particularly suited to the computation of orbits. Other multistep integration methods for first and second order ordinary differential equations are discussed as well. Accompanying it is code for computing the coefficients. This publication is posted with the permission of the American Astronautical Society.The Gauss-Jackson multi-step predictor-corrector method is widely used in numerical integration problems for astrodynamics and dynamical astronomy. The U.S. space surveillance centers have used an eighth-order Gauss-Jackson algorithm since the 1960s. In this paper, we explain the algorithm including a derivation from first principals and its relation to other multi-step integration methods. We also study its applicability to satellite orbits including its accuracy and stability

Accuracy and Speed Effects of Variable Step Integration for Orbit Determination and Propagation

In this paper the fixed step Gauss-Jackson method is compared to two variable step integrators. The first is the variable step, variable order Shampine-Gordon method. The second is s-integration, which may be considered an analytical step regulation. Speed tests are performed for orbit propagation with the integrators set to give equivalent accuracy. The integrators are also tested for orbit determination, to determine the speed benefit of the variable step methods. The tests give an indication of the types of orbits where variable step methods are more efficient than fixed step methods

A Variable-Step Double-Integration Multi-Step Integrator

See also the dissertation of Matt Berry at http://scholar.lib.vt.edu/theses/available/etd-04282004-071227/A variable-step double-integration multi-step integrator is derived using divided differences. The derivation is based upon the derivation of Shampine-Gordon, a single-integration method. Variable-step integrators are useful for propagating elliptical orbits, because larger steps can be taken near apogee. As a double-integration method, the integrator performs only one function evaluation per step, whereas Shampine-Gordon requires two evaluations per step, giving the integrator a significant speed advantage over Shampine-Gordon. Though several implementation issues remain, preliminary results show the integrator to be effective

Comparison of MSIS and Jacchia atmospheric density models for orbit determination and propagation

Two atmospheric density model families that are commonly chosen for orbit determination and propagation, Jacchia and MSIS, are compared for accuracy. The Jacchia 70 model, the MSISE-90 model, and the NRLMSISE-00 model may each be used to determine orbits over fitspans of several days and then to propagate forward. With observations kept over the propagation period, residuals may be computed and the accuracy of each model evaluated. We have performed this analysis for over 4000 cataloged satellites with perigee below 1000km for September-­October 1999, and the 60 HASDM calibration satellites with a large observation set for February 2001. The purpose of this study is to form a picture of the relative merits of the drag models in a comprehensive view, using all satellites in a manner consistent with the operational practice of US space surveillance centers. A further goal is to refine this knowledge to understand the orbital parameter regions where one of the models may be consistently superior

Effects of centrally acting ACE inhibitors on the rate of cognitive decline in dementia

Objectives: There is growing evidence that antihypertensive agents, particularly centrally acting ACE inhibitors (CACE-Is), which cross the blood–brain barrier, are associated with a reduced rate of cognitive decline. Given this, we compared the rates of cognitive decline in clinic patients with dementia receiving CACE-Is (CACE-I) with those not currently treated with CACE-Is (NoCACE-I), and with those who started CACE-Is, during their first 6 months of treatment (NewCACE-I). Design: Observational case–control study. Setting: 2 university hospital memory clinics. Participants: 817 patients diagnosed with Alzheimer's disease, vascular or mixed dementia. Of these, 361 with valid cognitive scores were included for analysis, 85 CACE-I and 276 NoCACE-I. Measurements: Patients were included if the baseline and end-point (standardised at 6 months apart) Standardised Mini-Mental State Examination (SMMSE) or Quick Mild Cognitive Impairment (Qmci) scores were available. Patients with comorbid depression or other dementia subtypes were excluded. The average 6-month rates of change in scores were compared between CACE-I, NoCACE-I and NewCACE-I patients. Results: When the rate of decline was compared between groups, there was a significant difference in the median, 6-month rate of decline in Qmci scores between CACE-I (1.8 points) and NoCACE-I (2.1 points) patients (p=0.049), with similar, non-significant changes in SMMSE. Median SMMSE scores improved by 1.2 points in the first 6 months of CACE treatment (NewCACE-I), compared to a 0.8 point decline for the CACE-I (p=0.003) group and a 1 point decline for the NoCACE-I (p=0.001) group over the same period. Multivariate analysis, controlling for baseline characteristics, showed significant differences in the rates of decline, in SMMSE, between the three groups, p=0.002. Conclusions: Cognitive scores may improve in the first 6 months after CACE-I treatment and use of CACE-Is is associated with a reduced rate of cognitive decline in patients with dementia