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

- 214 Downloads
- 1 Citations

## 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

## References

- Abramowitz, M., Stegun, I.A.: Handbook of Math. Functions With Formulas, Graphs, and Mathematical Tables, pp. 569–580, 649–650. Courier Dover Publications, Mineola (1972)Google Scholar
- Bate, R.R., Mueller, D.D., White, J.E.: Fundamentals of Astrodynamics, pp. 177–226. Dover Publications Inc., Mineola (1971)Google Scholar
- Beletsky, V.V.: Essays on the Motion of Celestial Bodies. Birkhäuser Verlag, Basel (2001)CrossRefMATHGoogle Scholar
- Biscani, F., Izzo, D.: Python Code for the Implementation of the Solution of the Stark Problem Via Weierstrass Elliptic and Related Functions. https://github.com/bluescarni/stark_weierstrass (2013). Accessed 01 Nov 2013
- Biscani, F., Izzo, D.: The Stark problem in the Weierstrassian formalism. Mon. Not. R. Astron. Soc.
**439**, 810–822 (2014). doi: 10.1093/mnras/stt2501 - Fenton, J.D., Gardiner-Garden, R.S.: Rapidly-convergent methods for evaluating elliptic integrals and theta and elliptic functions. J. Austral. Math. Soc.
**24**, 47–58 (1982)CrossRefMATHMathSciNetGoogle Scholar - Fukushima, T.: Precise and fast computation of a general incomplete elliptic integral of third kind by half and double argument transformations. J. Comput. Appl. Math.
**236**, 1961–1975 (2012)CrossRefMATHMathSciNetGoogle Scholar - Fukushima, T.: Fast computation of a general complete elliptic integral of third kind by half and double argument transformations. J. Comput. Appl. Math.
**253**, 142–157 (2013a)CrossRefMATHMathSciNetGoogle Scholar - Fukushima, T.: Precise and fast computation of Jacobian elliptic functions by conditional duplication. Numer. Math. (2013b). doi: 10.1007/s00211-012-0498-0
- Fukushima, T.: Fukushima, t. personal researchgate web page. https://www.researchgate.net/profile/Toshio_Fukushima (2014). Accessed 02 Aug 2014
- Fukushima, T., Ishizaki, H.: Numerical computation of incomplete elliptic integrals of a general form. Celest. Mech. Dyn. Astron.
**59**, 237–251 (1994)ADSCrossRefMATHMathSciNetGoogle Scholar - Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press, New York (2007)MATHGoogle Scholar
- Isayev, Y., Kunitsyn, A.: To the problem of satellite’s perturbed motion under the influence of solar radiation pressure. Celest. Mech. Dyn. Astron.
**6**, 44–51 (1972)CrossRefGoogle Scholar - Kirchgraber, U.: A problem of orbital dynamics, which is separable in ks-variables. Celest. Mech. Dyn. Astron.
**4**, 340–347 (1971)CrossRefMATHGoogle Scholar - Lagrange, JL.: Mécanique Analytique. Courcier, Paris (1788)Google Scholar
- Lantoine, G.: A Methodology for Robust Optimization of Low-Thrust Trajectories in Multi-Body Environments. Ph.D. thesis, Georgia Institute of Technology, pp. 125–181 (2010)Google Scholar
- Lantoine, G., Russell, R.: The Stark model: an exact, closed-form approach to low-thrust trajectory optimization. In: 21st International Symposium on Space Flight Dynamics (2009)Google Scholar
- Lantoine, G., Russell, R.P.: Complete closed-form solutions of the Stark problem. Celest. Mech. Dyn. Astron.
**109**(4), 333–366 (2011). doi: 10.1007/s10569-010-9331-1 - Nacozy, P.: A discussion of time transformations and local truncation errors. Celest. Mech.
**13**(4), 495–501 (1976)ADSCrossRefGoogle Scholar - Pellegrini, E., Russell, R.P., Vittaldev, V.: F and g Stark and Kepler series. http://russell.ae.utexas.edu/index_files/fgstark.htm (2013). Accessed 01 Nov 2013
- Pellegrini, E., Russell, R.P., Vittaldev, V.: F and g Tylor series solutions to the Stark and Kepler problems with Sundman transformations. Celest. Mech. Dyn. Astron., 1–24 (2014). doi: 10.1007/s10569-014-9538-7
- Petropoulos, A.E.: Refinements to the q-law for low-thrust orbit transfers. In: AAS/AIAA Space Flight Mechanical Conference, Copper Mountain CO (2005)Google Scholar
- Radhakrishnan, K., Hindmarsh, A.: Description and use of lsode, the livermore solver for ordinary differential equations. NASA Ref. Publication 1327, NASA (1993)Google Scholar
- Rufer, D.: Trajectory optimization by making use of the closed solution of constant thrust-acceleration motion. Celest. Mech.
**14**, 91–103 (1976)Google Scholar - Sims, J.A., Flanagan, S.N.: Preliminary design of low-thrust interplanetary missions. In: AAS/AIAA Astrodynamics Specialist Conference and Exhibit, Girdwood, AK (1999)Google Scholar
- Sims, J.A., Finlayson, P., Rinderle, E., Vavrina, M., Kowalkowski, T.: Implementation of a low-thrust trajectory optimization algorithm for preliminary design. In: AAS/AIAA Astrodynamics Specialist Conference and Exhibit, Keystone, CO (2006)Google Scholar
- Stark, J.: Beobachtungen ber den effekt des elektrischen feldes auf spektrallinien. i. quereffekt. Annalen der Physik
**43**, 965–983 (1914)Google Scholar - Sundman, K.F.: Mémoire sur le problème des trois corps. Acta Math.
**36**, 105–179 (1912)CrossRefMATHMathSciNetGoogle Scholar - Yam, C.H., Longuski, J.M.: Reduced parameterization for optimization of low-thrust gravity-assist trajectories: case studies. In: AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Keystone, CO (2006)Google Scholar