Highlights Revisiting Lambert’s problem and the Lagrangian form of the elliptic time of flight equation. A Lambert method was developed to first and second orders (for comparison) and solved 28 test cases, including difficult ones. The method is numerical but relies on analytical derivatives. It supports 180° transfers, unlike most Lambert solvers. Results indicate that the speed of convergence depends more on the accuracy of the first guess than the order of the method.

    Abstract In the two-body model, time of flight between two positions can be expressed as a single-variable function and a variety of formulations exist. Lambert’s problem can be solved by inverting such a function. In this article, a method which inverts Lagrange’s flight time equation and supports the problematic 180 ° transfer is proposed. This method relies on a Householder algorithm of variable order. However, unlike other iterative methods, it is semi-analytical in the sense that flight time functions are derived analytically to second order vs. first order finite differences. The author investigated the profile of Lagrange’s elliptic flight time equation and its derivatives with a special focus on their significance to the behaviour of the proposed method and the stated goal of guaranteed convergence. Possible numerical deficiencies were identified and dealt with. As a test, 28 scenarios of variable difficulty were designed to cover a wide variety of geometries. The context of this research being the orbit determination of artificial satellites and debris, the scenarios are representative of typical such objects in Low-Earth, Geostationary and Geostationary Transfer Orbits. An analysis of the computational impact of the quality of the initial guess vs. that of the order of the method was also done, providing clues for further research and optimisations (e.g. asteroids, long period comets, multi-revolution cases). The results indicate fast to very fast convergence in all test cases, they validate the numerical safeguards and also give a quantitative assessment of the importance of the initial guess.


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    Title :

    On Lambert’s problem and the elliptic time of flight equation: A simple semi-analytical inversion method


    Contributors:

    Published in:

    Advances in Space Research ; 53 , 5 ; 890-898


    Publication date :

    2013-12-26


    Size :

    9 pages




    Type of media :

    Article (Journal)


    Type of material :

    Electronic Resource


    Language :

    English





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