# Comparison of three Stark problem solution techniques for the bounded case

## Abstract

Three methods of obtaining solutions to the Stark problem—one developed by Lantoine and Russell using Jacobi elliptic and related functions, one developed by Biscani and Izzo using Weierstrass elliptic and related functions, and one developed by Pellegrini, Russell, and Vittaldev using \(F\) and \(G\) Taylor series extended to the Stark problem—are compared qualitatively and quantitatively for the bounded motion case. For consistency with existing available code for the series solution, Fortran routines of the Lantoine method and Biscani method are newly implemented and made available. For these implementations, the Lantoine formulation is found to be more efficient than the Biscani formulation in the propagation of a single trajectory segment. However, for applications for which acceptable accuracy may be achieved by orders up to 16, the Pellegrini series solution is shown to be more efficient than either analytical method. The three methods are also compared in the propagation of sequentially connected trajectory segments in a low-thrust orbital transfer maneuver. Separate tests are conducted for discretizations between 8 and 96 segments per orbit. For the series solution, the interaction between order and step size leads to computation times that are nearly invariable to discretization for a given truncation error tolerance over the tested range of discretizations. This finding makes the series solution particularly attractive for mission design applications where problems may require both coarse and fine discretizations. Example applications include the modeling of low-thrust propulsion and time-varying perturbations—problems for which the efficient propagation of relatively short Stark segments is paramount because the disturbing acceleration generally varies continuously.

## Keywords

Stark problem Jacobi functions Weierstrass functions F and G Taylor series Trajectory propagation Low-thrust transfer Lantoine method Biscani method Pellegrini method## Notes

### Acknowledgments

The authors would like to thank Gregory Lantoine and Nicholas Bradley for their assistance in the implementation of the Lantoine formulation of the Stark problem and Etienne Pellegrini for his assistance in implementing the series formulation of the Stark problem. The authors also thank the referees for comments and suggestions that we believe improved the quality of this paper.

## Supplementary material

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