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  • 1
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    An approach for finding long period elliptical orbits for precursor SEI missions (1993)
    In:  CASI
    Details
    Publication Date: 2013-08-31
    Description: Precursors for Solar System Exploration Initiative (SEI) missions may require long period elliptical orbits about a planet. These orbits will typically have periods on the order of tens to hundreds of days. Some potential uses for these orbits may include the following: studying the effects of galactic cosmic radiation, parking orbits for engineering and operational test of systems, and ferrying orbits between libration points and low altitude orbits. This report presents an approach that can be used to find these orbits. The approach consists of three major steps. First, it uses a restricted three-body targeting algorithm to determine the initial conditions which satisfy certain desired final conditions in a system of two massive primaries. Then the initial conditions are transformed to an inertial coordinate system for use by a special perturbation method. Finally, using the special perturbation method, other perturbations (e.g., sun third body and solar radiation pressure) can be easily incorporated to determine their effects on the nominal trajectory. An algorithm potentially suitable for on-board guidance will also be discussed. This algorithm uses an analytic method relying on Chebyshev polynomials to compute the desired position and velocity of the satellite as a function of time. Together with navigation updates, this algorithm can be implemented to predict the size and timing for AV corrections.
    Keywords: ASTRODYNAMICS
    Type: NASA. Goddard Space Flight Center, Flight Mechanics(Estimation Theory Symposium, 1992; p 381-394
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  • 2
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    A general time element for orbit integration in Cartesian coordinates (1981)
    In:  Other Sources
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    Publication Date: 2011-08-18
    Description: Two techniques are discussed for increasing the accuracy of the numerical integration of eccentric orbits in Cartesian coordinates. One involves the use of an independent variable different from time; this increases the efficiency of the numerical integration. The other uses a time element, which reduces the in-track error. A general expression is given of a time element valid for an arbitrary independent variable. It is pointed out that this time element makes it possible to switch the independent variable merely by applying a scaling factor; there is no need to change the differential equations of the motion. Eccentric, true, and elliptic anomalies are used as independent variables in the case of a transfer orbit for a geosynchronous orbit. The elliptic anomaly is shown to perform much better than the other classical anomalies.
    Keywords: ASTRODYNAMICS
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  • 3
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    Canonical orbital elements in terms of an arbitrary independent variable (1981)
    In:  Other Sources
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    Publication Date: 2011-08-18
    Description: Within the framework of the Hamiltonian mechanics in the extended phase space, a set of canonical elements of the Delaunay type is developed in terms of an arbitary independent angular variable. Application to the four classical anomalies - eccentric, true, elliptic, and mean - is presented. Particular attention is given to the generalized time equation and its conjugate energy equation.
    Keywords: ASTRODYNAMICS
    Type: Celestial Mechanics; 23; Feb. 198
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  • 4
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    Analytical satellite theory in extended phase space (1980)
    In:  Other Sources
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    Publication Date: 2011-08-17
    Description: It is noted that a satellite theory, based on extended phase space and on the true anomaly, was introduced by Scheifele (1970). In the present paper a simple canonical transformation is shown that makes the transition from the classical Delaunay elements to the Scheifele variables. It is stressed that neither spherical coordinates nor Hamilton-Jacobi theory is used. Finally, attention is given to the meaning of the new variables, especially the use of the true anomaly as one of the variables.
    Keywords: ASTRODYNAMICS
    Type: Celestial Mechanics; 21; May 1980
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  • 5
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    The development of the Poincare-similar elements with eccentric anomaly as the independent variable (1976)
    In:  Other Sources
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    Publication Date: 2011-08-16
    Description: A new set of element differential equations for the perturbed two-body motion is derived. The elements are canonical and are similar to the classical canonical Poincare elements, which have time as the independent variable. The phase space is extended by introducing the total energy and time as canonically conjugated variables. The new independent variable is, to within an additive constant, the eccentric anomaly. These elements are compared to the Kustaanheimo-Stiefel (KS) element differential equations, which also have the eccentric anomaly as the independent variable. For several numerical examples, the accuracy and stability of the new set are equal to those of the KS solution. This comparable accuracy result can probably be attributed to the fact that both sets have the same time element and very similar energy elements. The new set has only 8 elements, compared to 10 elements for the KS set. Both sets are free from singularities due to vanishing eccentricity and inclination.
    Keywords: ASTRODYNAMICS
    Type: Celestial Mechanics; 13; May 1976
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  • 6
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    Propagation of local errors in the solutions of the differential equations for orbital elements (1982)
    In:  Other Sources
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    Publication Date: 2011-08-18
    Description: The propagation of errors in the solutions of the differential equations for the orbital elements of perturbed two-body motion is investigated. It is shown that the error in the time-element grows linearly for differential equations for orbital elements when only perturbations are present on the right-hand side, cubically for formulations which have a two-body term on the right-hand side, and linearly for formulations based upon extended phase space Hamiltonians.
    Keywords: ASTRODYNAMICS
    Type: Celestial Mechanics; 27; June 198
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  • 7
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    Error propagation in the numerical solutions of the differential equations of orbital mechanics (1982)
    In:  Other Sources
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    Publication Date: 2011-08-18
    Description: The relationship between the eigenvalues of the linearized differential equations of orbital mechanics and the stability characteristics of numerical methods is presented. It is shown that the Cowell, Encke, and Encke formulation with an independent variable related to the eccentric anomaly all have a real positive eigenvalue when linearized about the initial conditions. The real positive eigenvalue causes an amplification of the error of the solution when used in conjunction with a numerical integration method. In contrast an element formulation has zero eigenvalues and is numerically stable.
    Keywords: ASTRODYNAMICS
    Type: Celestial Mechanics; 27; May 1982
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  • 8
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    The elliptic anomaly (1980)
    In:  CASI
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    Publication Date: 2019-06-27
    Description: An independent variable different from the time for elliptic orbit integration is used. Such a time transformation provides an analytical step-size regulation along the orbit. An intermediate anomaly (an anomaly intermediate between the eccentric and the true anomaly) is suggested for optimum performances. A particular case of an intermediate anomaly (the elliptic anomaly) is defined, and its relation with the other anomalies is developed.
    Keywords: ASTRODYNAMICS
    Type: NASA-TM-58228
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  • 9
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    An Analytical Singularity-Free Solution to the J2 Perturbation Problem (1979)
    In:  CASI
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    Publication Date: 2019-06-27
    Description: The development of a singularity-free solution of the J2 problem in satellite theory is presented. The procedure resembles that of Lyndane who rederives Brouwer's satellite theory using Poincare elements. A comparable procedure is used in this report in which the satellite theory of Scheifele, who used elements similar to the Delaunay elements but in the extended phase space, is rederived using Poincare elements also in the extended phase space. Only the short-period effects due to J2 are included.
    Keywords: ASTRODYNAMICS
    Type: NASA-TM-58221 , JSC-13128
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  • 10
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    Computation of orbits using total energy (1988)
    In:  CASI
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    Publication Date: 2019-06-28
    Description: The computation of orbits can be done more efficiently by the use of any of several new formulations of the perturbed two body problem which consider the total energy of the orbital system as one of the dependent variables. The total energy is the osculating two body energy plus the potential energy due to perturbing masses. The use of the total energy as the dependent variable instead of the two body energy is a relatively new idea. The advantage of using total energy arises from the fact that the more perturbing potential energy that is accounted for in the total energy variable, the more nearly constant is the total energy. In fact, except for dissipative forces such as drag, the only reason for the total energy not being constant is the rotation or revolution of the perturbing mass. This near constancy of the total energy has the effect of inhibiting error growth during numerical solution. This paper will present the results of an application of total energy formulation to the problem of the precise computation of orbits.
    Keywords: ASTRODYNAMICS
    Type: NASA, Goddard Space Flight Center, Flight Mechanics(Estimation Theory Symposium 1988; p 153-165
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