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F and G Taylor series solutions to the Stark and Kepler problems with Sundman transformations

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Abstract

The classic \(F\) and \(G\) Taylor series of Keplerian motion are extended to solve the Stark problem and to use the generalized Sundman transformation. Exact recursion formulas for the series coefficients are derived, and the method is implemented to high order via a symbolic manipulator. The results lead to fast and accurate propagation models with efficient discretizations. The new \(F\) and \(G\) Stark series solutions are compared to the Modern Taylor Series (MTS) and 8th order Runge–Kutta–Fehlberg (RKF8) solutions. In terms of runtime, the \(F\) and \(G\) approach is shown to compare favorably to the MTS method up to order 20, and both Taylor series methods enjoy approximate order of magnitude speedups compared to RKF8 implementations. Actual runtime is shown to vary with eccentricity, perturbation size, prescribed accuracy, and the Sundman power law. The method and results are valid for both the Stark and the Kepler problems. The effects of the generalized Sundman transformation on the accuracy of the propagation are analyzed. The Taylor series solutions are shown to be exceptionally efficient when the unity power law from the classic Sundman transformation is applied. An example low-thrust trajectory propagation demonstrates the utility of the \(F\) and \(G\) Stark series solutions.

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Notes

  1. Subject to the hardware and software specifications and the implementation described in Sect. 4.

  2. An up-to-date version of the code can be downloaded from http://russell.ae.utexas.edu/index_files/fgstark.htm.

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Acknowledgments

This material is based upon work supported in part by the Space Vehicles Directorate of the Air Force Research Laboratory under Contract No. FA9453-13-C-0205. The authors are grateful for their interest and support.

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Authors and Affiliations

  1. Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX, USA

    Etienne Pellegrini, Ryan P. Russell & Vivek Vittaldev

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  1. Etienne Pellegrini
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  2. Ryan P. Russell
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  3. Vivek Vittaldev
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Correspondence to Etienne Pellegrini.

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Appendix: Algorithm used for the performance comparison of all integrators applied to an example low-thrust orbital transfer

Appendix: Algorithm used for the performance comparison of all integrators applied to an example low-thrust orbital transfer

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Pellegrini, E., Russell, R.P. & Vittaldev, V. F and G Taylor series solutions to the Stark and Kepler problems with Sundman transformations. Celest Mech Dyn Astr 118, 355–378 (2014). https://doi.org/10.1007/s10569-014-9538-7

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