The Journal of the Astronautical Sciences

, Volume 56, Issue 3, pp 311–324 | Cite as

Fast design of repeat ground track orbits in high-fidelity geopotentials

  • Martin Lara
  • Ryan P. Russell


The existence of families of periodic, repeat ground track orbits in full geopotentials is demonstrated. The basic families are made of almost circular orbits except in the vicinity of the critical inclination (63.4/116.6 deg), where the eccentricity of the repeat orbits grows for almost fixed inclination. Computation of specific repeat ground track orbits for mission design can be automated providing the nominal solution in a fast, straightforward way. We illustrate this with the computation of the TOPEX nominal orbit in a 140 × 140 truncation of the GRACE Gravity Model.


Periodic Orbit Gravity Field Semimajor Axis Jacobi Constant Differential Correction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    LARA, M. “Searching for Repeating Ground Track Orbits: A Systematic Approach,” The Journal of the Astronautical Sciences, Vol. 47, 1999, pp. 177–188.Google Scholar
  2. [2]
    LARA, M. “Repeat Ground Track Orbits of the Earth Tesseral Problem as Bifurcations of the Equatorial Family of Periodic Orbits,” Celestial Mechanics and Dynamical Astronomy, Vol. 86, No. 2, 2003, pp. 143–162.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    LARA, M. and RUSSELL, R. P. “On the Computation of a Science Orbit About Europa,” Journal of Guidance, Control, and Dynamics, Vol. 30, No. 1, 2007, pp. 259–263.MathSciNetCrossRefGoogle Scholar
  4. [4]
    RUSSELL, R. P. and LARA, M. “Long Lifetime Lunar Repeat Ground Track Orbits,” Journal of Guidance, Control, and Dynamics, Vol. 30, No. 4, 2007, pp. 982–993.MathSciNetCrossRefGoogle Scholar
  5. [5]
    SEIDELMANN, P. K., ARCHINAL, B. A., AHEARN, M. F., CONRAD, A., CONSOLMAGNO, G. J., HESTROFFER, D., HILTON, J. L., KRASINSKY, G. A., NEUMANN, G., OBERST, J., STOOKE, P., TEDESCO, E. F., THOLEN, D. J., THOMAS, P. C., and WILLIAMS, I. P. “Report of the IAU/IAG Working Group on Cartographic Coordinates and Rotational Elements: 2006,” Celestial Mechanics and Dynamical Astronomy, Vol. 98, No. 3, 2007, DOI 10.1007/s10569-007-9072-y, pp. 155–180.zbMATHCrossRefGoogle Scholar
  6. [6]
    IRIGOYEN, M. and SIMÓ, C. “Non Integrability of the Problem,” Celestial Mechanics and Dynamical Astronomy, Vol. 55, No. 3, 1993, pp. 281–287.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    DEPRIT, A. and Henrard, J. “Construction of Orbits Asymptotic to a Periodic Orbit,” The Astronomical Journal, Vol. 72, No. 2, 1969, pp. 308–316.CrossRefGoogle Scholar
  8. [8]
    TAPLEY, B., RIES, J., BETTADPUR, S., CHAMBERS, D., CHENG, M., CONDI, F., GUNTER, B., KANG, Z., NAGEL, P., PASTOR, R., PEKKER, T., POOLE, S., and WANG, F. “GGM02-An Improved Earth Gravity Field Model from GRACE,” Journal of Geodesy, Vol. 79, 2005, DOI 10.1007/s00190-005-0480-z, pp. 467–478.CrossRefGoogle Scholar
  9. [9]
    LARA, M. and PELÁEZ, J. “On the Numerical Continuation of Periodic Orbits: An Intrinsic, 3-Dimensional, Differential, Predictor-Corrector Algorithm,” Astronomy and Astrophysics, Vol. 389, Feb. 2002, pp. 692–701.zbMATHCrossRefGoogle Scholar
  10. [10]
    HENÒN, M. “Exploration Numérique du Problème Restreint. II.—Masses égales, stabilité des orbites périodiques,” Annales d’Astrophysique, Vol. 28, No. 2, 1965, pp. 992–1007.Google Scholar
  11. [11]
    VALLADO, D.A. Fundamentals of Astrodynamics and Applications, 2nd edition, 2004, Space Technology Library, Microcosm Press & Kluwer Academic Publishers, pp. 792 ff.Google Scholar
  12. [12]
    LARA, M. Sadsam: a Software Assistant for Designing Satellite Missions, Report CNES num. DTS/MPI/MS/MN/99-053, 1999, 75 pages.Google Scholar
  13. [13]
    FRAUENHOLZ, R. B., BHAT, R. S., and SHAPIRO, B. E. “Analysis of the TOPEX/Poseidon Operational Orbit: Observed Variations and Why,” Journal of Spacecraft and Rockets, Vol. 35, No. 2, 1998, pp. 212–224.CrossRefGoogle Scholar
  14. [14]
    BROUCKE, R. “Stability of Periodic Orbits in the Elliptic, Restricted Three-Body Problem,” AIAA Journal, Vol. 7, 1969, pp. 1003–1009.zbMATHCrossRefGoogle Scholar
  15. [15]
    LARA, M. and SAN JUAN, J. F. “Dynamic Behavior of an Orbiter Around Europa,” Journal of Guidance, Control and Dynamics, Vol. 28, No. 2, 2005, pp. 291–297.CrossRefGoogle Scholar
  16. [16]
    BROUCKE, R. “Numerical Integration of Periodic Orbits in the Main Problem of Artificial Satellite Theory,” Celestial Mechanics and Dynamical Astronomy, Vol. 58, No. 2, 1994, pp. 99–123.MathSciNetCrossRefGoogle Scholar
  17. [17]
    BROUCKE, R. “Periodic Collision Orbits in the Elliptic Restricted Three-Body Problem,” Celestial Mechanics and Dynamical Astronomy, Vol. 3, 1971, pp. 461–477.zbMATHGoogle Scholar
  18. [18]
    COFFEY, S., DEPRIT, A., and DEPRIT, E. “Frozen Orbits for Satellites Close to an Earth-Like Planet,” Celestial Mechanics and Dynamical Astronomy, Vol. 59, No. 1, 1994, pp. 37–72.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    PINES, S. “Uniform Representation of the Gravitational Potential and its Derivatives,” AIAA Journal, Vol. 11, Nov. 1973, pp. 1508–1511.zbMATHCrossRefGoogle Scholar
  20. [20]
    LUNDBERG, J. B. and SCHUTZ, B. E. “Recursion Formulas of Legendre Functions for use with Nonsingular Geopotential Models,” Journal of Guidance, Vol. 11, No. 1, 1988, pp. 31–38.zbMATHCrossRefGoogle Scholar
  21. [21]
    HAIRER, E., NØRSETT, S. P., and WANNER, G. Solving Ordinary Differential Equations. Nonstiff Problems, 2nd edition, 1993, Springer Series in Computational Mathematics, Vol. 8, pp. 181–185.zbMATHGoogle Scholar
  22. [22]
    BOND, V. “Error Propagation in the Numerical Solution of the Differential Equations of Orbital Mechanics,” Celestial Mechanics, Vol. 27, No. 1, 1982, pp. 65–77.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    LAWSON, C. L. and HANSON, R. J. Solving Ordinary Least Squares Problems, 1974, Prentice-Hall.Google Scholar
  24. [24]
    Math77 Reference Manual, edited by Lisa K. Jones and Laurie Seaton, Language Systems Corporation, Sterling, Va., 1994.Google Scholar

Copyright information

© American Astronautical Society, Inc. 2008

Authors and Affiliations

  1. 1.Ephemeris SectionReal Observatorio de la ArmadaSan FernandoSpain
  2. 2.School of Aerospace EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations