Analytical Solutions for Extremal Space Trajectories by Dilmurat M. Azimov

Author:Dilmurat M. Azimov
Language: eng
Format: epub
ISBN: 9780128140598
Publisher: Elsevier B.V.
Published: 2017-08-23T16:00:00+00:00

One can also verify that, as it will be shown in the next subsection, the other solutions for the spirals can be obtained from (4.7), (4.13)–(4.15).

Remark 2

In Ref. [152], the Lawden spirals represent the solutions valid in the case of free time and minimization of the fuel mass. It is noted that there exists a possibility that the corresponding trajectory may be a solution to the problem of optimal escape from a circular orbit. However, in this specific case, from the transversality condition (see (4.140)) it follows that , as the functional of the problem does not depend on the final polar angle. Using (4.135) we obtain , and this leads to the degeneration of the thrust arcs. Consequently, in Ref. [152] the IT arcs, that is the Lawden spirals can not be included into an optimal trajectory of escape from a circular orbit.

Consequently, the spirals can be the solutions only in the case when the final polar angle is given or the functional of the problem explicitly depends on the final polar angle. In particular, if the minimization of the characteristic velocity is considered, and there is no constraint on the angular distance, the Lawden spirals can not serve as the IT arc solutions to the problem.

In summary, the following comments can be made regarding the optimality and applicability of the Lawden spirals. According to the variation problem, the IT arcs have to satisfy the canonical equations, the boundary conditions, the transversality conditions, and the conditions . The latter conditions represent the conditions of existence and optimality of the thrust arcs.

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