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  • 1
    Electronic Resource
    Electronic Resource
    The uniform, regular differential equations of the KS transformed perturbed two-body problem (1974)
    Springer
    Celestial mechanics and dynamical astronomy 10 (1974), S. 303-318 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The Newtonian differential equations of motion for the two-body problem can be transformed into four, linear, harmonic oscillator equations by simultaneously applying the regularizing time transformation dt/ds=r and the Kustaanheimo-Stiefel (KS) coordinate transformation. The time transformation changes the independent variable from time to a new variables, and the KS transformation transforms the position and velocity vectors from Cartesian space into a four-dimensional space. This paper presents the derivation of uniform, regular equations for the perturbed twobody problem in the four-dimensional space. The variation of parameters technique is used to develop expressions for the derivatives of ten elements (which are constants in the unperturbed motion) for the general case that includes both perturbations which can arise from a potential and perturbations which cannot be derived from a potential. These element differential equations are slightly modified by introducing two additional elements for the time to further improve long term stability of numerical integration.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01586860
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  • 2
    Electronic Resource
    Electronic Resource
    The development of the Poincaré-similar elements with eccentric anomaly as the independent variable (1976)
    Springer
    Celestial mechanics and dynamical astronomy 13 (1976), S. 287-311 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract 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 Poincaré 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01228648
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  • 3
    Electronic Resource
    Electronic Resource
    The development of the Poincaré-similar elements with eccentric anomaly as the independent variable (1976)
    Springer
    Celestial mechanics and dynamical astronomy 14 (1976), S. 333-333 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract A new set of element differential equations for the perturbed two-body motions is derived. The elements are canonical and are similar to the classical canonical Poincaré 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01228518
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  • 4
    Electronic Resource
    Electronic Resource
    A transformation of the two-body problem (1985)
    Springer
    Celestial mechanics and dynamical astronomy 35 (1985), S. 1-7 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract A transformation of the differential equations of motion of the two-body problem in the spherical coordinates to oscillator form is derived. It is shown that the independent variable transformation dt/ds=r2 is a transformation which makes the oscillator form possible.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01229108
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  • 5
    Electronic Resource
    Electronic Resource
    Canonical orbital elements in terms of an arbitrary independent variable (1981)
    Springer
    Celestial mechanics and dynamical astronomy 23 (1981), S. 159-172 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract 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 arbitrary 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01229551
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  • 6
    Electronic Resource
    Electronic Resource
    Error propagation in the numerical solutions of the differential equations of orbital mechanics (1982)
    Springer
    Celestial mechanics and dynamical astronomy 27 (1982), S. 65-77 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The relationship between the eigen values 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 eigen value when linearized about the initial conditions. The real positive eigen value 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 eigen values and is numerically stable.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01228951
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  • 7
    Electronic Resource
    Electronic Resource
    Propagation of local errors in the solutions of the differential equations for orbital elements (1982)
    Springer
    Celestial mechanics and dynamical astronomy 27 (1982), S. 203-210 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01271694
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  • 8
    Electronic Resource
    Electronic Resource
    The classical solution of a definite integral occurring in the satellite theory in the extended phase space (1979)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 30 (1979), S. 159-166 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Description / Table of Contents: Zusammenfassung Das bestimmte Integral $$\frac{1}{{2\pi }}\int_0^{2\pi } {\cos } [m\phi + nv(E - \phi - e\sin E]d\phi $$ wird durch wiederholte Anwendung einer Identität über Besselfunktionen berechnet. Dann werden die klassischen Reihen für die Besselfunktionen benützt, um die Lösung nach der Exzentrizität zu entwickeln.
    Notes: Abstract The definite integral $$\frac{1}{{2\pi }}\int_0^{2\pi } {\cos } [m\phi + nv(E - \phi - e\sin E]d\phi $$ is solved by repeated application of a Bessel function identity. The classical series for the Bessel functions are then used to expand the solution with respect to the eccentricity.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01601930
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  • 9
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    Propagation of local errors in the solutions of the differential equations for orbital elements (1982)
    Springer
    Details
    Publication Date: 1982-06-01
    Print ISSN: 0923-2958
    Electronic ISSN: 1572-9478
    Topics: Physics
    Published by Springer
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  • 10
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    Error propagation in the numerical solutions of the differential equations of orbital mechanics (1982)
    Springer
    Details
    Publication Date: 1982-05-01
    Print ISSN: 0923-2958
    Electronic ISSN: 1572-9478
    Topics: Physics
    Published by Springer
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