Celestial Mechanics and Dynamical Astronomy

, Volume 116, Issue 3, pp 213–227 | Cite as

Motion near frozen orbits as a means for mitigating satellite relative drift

  • P. Gurfil
  • M. Lara
Original Article


Generally, any initially-close satellites—chief and deputy—moving on orbits with slightly different orbital elements, will depart each other on locally unbounded relative trajectories. Thus, constraints on the initial conditions must be imposed to mitigate the chief-deputy mutual departure. In this paper, it is analytically proven that choosing the chief’s orbit to be a frozen orbit can mitigate the natural relative drift of the satellites. Using mean orbital element variations, it is proven that if the chief’s orbit is frozen, then the mean differential eccentricity is periodic, leading to a periodic variation of the differential mean argument of latitude. On the other hand, if the chief’s orbit is non-frozen, a secular growth in the differential mean argument of latitude leads to a concomitant along-track separation of the deputy from the chief, thereby considerably increasing the relative distance evolution over time. Long-term orbital simulation results indicate that the effect of choosing a frozen orbit vis-à-vis a non-frozen orbit can reduce the relative distance drift by hundreds of meters per day.


Cluster flight Orbital mechanics Frozen orbits  Zonal harmonics 



This work was supported by the European Research Council Starting Independent Researcher Grant # 278231: Flight Algorithms for Disaggregated Space Architectures (FADER).


  1. Alfriend, K., Vadali, S., Gurfil, P., How, J., Breger, L.: Spacecraft Formation Flying: Dynamics, Control and Navigation. Elseiver, Oxford, UK, chapter VI (2010)Google Scholar
  2. Battin, R.H.: An Introduction to the Mathematics and Methods of Astrodynamics. AIAA, Reston (1999)zbMATHCrossRefGoogle Scholar
  3. Broucke, R.A.: Numerical integration of periodic orbits in the main problem of artificial satellite theory. Celest. Mech. Dyn. Astron. 58(2), 99–123 (1994)MathSciNetADSCrossRefGoogle Scholar
  4. Brouwer, D.: Solution of the problem of artificial satellite theory without drag. Astron. J. 64, 378–397 (1959)MathSciNetADSCrossRefGoogle Scholar
  5. Coffey, S.L., Deprit, A., Miller, B.R.: The critical inclination in artificial satellite theory. Celest. Mech. 39(4), 365–406 (1986)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. Coffey, S.L., Deprit, A., Deprit, E.: Frozen orbits for satellites close to an earth-like planet. Celest. Mech. Dyn. Astron. 59(1), 37–72 (1994)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. Cushman, R.: Reduction, Brouwer’s Hamiltonian and the critical inclination. Celest. Mech. 31(4):401–429; errata: 33(3), 1984, p. 297 (1983)Google Scholar
  8. Cutting, E., Frautnick, J., Born, G.: Orbit analysis for SEASAT-A. J. Astronaut. Sci. 26, 315–342 (1978)ADSGoogle Scholar
  9. Davidz, H.L.: Use of Near-frozen Orbits for Satellite Formation Flying. Thesis, University of Cincinnati, M.Sc (2001)Google Scholar
  10. Deprit, A.: The elimination of the parallax in satellite theory. Celest. Mech. 24, 111–153 (1981)MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. Gim, D.W., Alfriend, K.T.: State transition matrix of relative motion for the perturbed noncircular reference orbit. J Guid. Control Dyn. 26(6), 956–971 (2003)CrossRefGoogle Scholar
  12. Gim, D.W., Alfriend, K.T.: Satellite relative motion using differential equinoctial elements. Celest. Mech. Dyn. Astron. 92(4), 295–336 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. Gurfil, P.: Effect of equinoctial precession on geosynchronous earth satellites. J. Guid. Control Dyn. 30(1), 237–247 (2007)ADSCrossRefGoogle Scholar
  14. Gurfil, P., Herscovitz, J., Pariente, M.: The SAMSON project—cluster flight and geolocation with three autonomous nano-satellites. In: 26th AIAA/USU Conference on Small Satellites (2012)Google Scholar
  15. Kasdin, N.J., Gurfil, P., Kolemen, E.: Canonical modelling of relative spacecraft motion via epicyclic orbital elements. Celest. Mech. Dyn. Astron. 92(4), 337–370 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. Kaula, W.M.: Theory of Satellite Geodesy. Blaisdell Publishing Company, Waltham, Massachusetts. Reprinted by Dover Publications, Mineola, NY, 2000 (1966)Google Scholar
  17. Kozai, Y.: Motion of a lunar orbiter. Publ. Astron. Soc. Jpn. 15, 301–312 (1963)ADSGoogle Scholar
  18. Lara, M.: SADSaM: A Software Assistant for Designing SAtellite Mission. CNES report no. DTS/MPI/MS/MN/99-053, May 1999, 75 pp. (1999a)Google Scholar
  19. Lara, M.: Searching for repeating ground track orbits: a systematic approach. J Astronaut. Sci. 47(3–4), 177–188 (1999b)Google Scholar
  20. Lara, M.: Repeat ground track orbits of the earth tesseral problem as bifurcations of the equatorial family of periodic orbits. Celest. Mech. Dyn. Astron. 86(2), 143–162 (2003)MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. Lara, M., Gurfil, P.: Integrable approximation of \(J_2\)-perturbed relative orbits. Celest. Mech. Dyn. Astron. 114(3), 229–254 (2012)MathSciNetADSCrossRefGoogle Scholar
  22. Lara, M., Russell, R.P.: Fast design of repeat ground track orbits in high-fidelity geopotentials. J Astronaut. Sci. 56(3), 311–324 (2008)CrossRefGoogle Scholar
  23. Lara, M., Deprit, A., Elipe, A.: Numerical continuation of families of frozen orbits in the zonal problem of artificial satellite theory. Celest. Mech. Dyn. Astron. 62(2), 167–181 (1995)MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. Martinusi, V., Gurfil, P.: Solutions and periodicity of satellite relative motion under even zonal harmonics perturbations. Celest. Mech. Dyn. Astron. 111(4), 387–414 (2011)MathSciNetADSCrossRefGoogle Scholar
  25. McLaughlin, C., Sabol, C., Swank, A., Burns, R., Luu, K.: Modeling relative position, relative velocity, and range rate for formation flying. Adv. Astronaut. Sci. 109(3), 2165–2186 (2002)Google Scholar
  26. Roh, K.M., Luehr, H., Rothacher, M., Park, S.Y.: Investigating suitable orbits for the swarm constellation mission—the frozen orbit. Aerosp. Sci. Technol. 13(1), 49–58 (2009)CrossRefGoogle Scholar
  27. Rosborough, G., Ocampo, C.: Influence of higher degree zonals on the frozen orbit geometry. In: Astrodynamics 1991, vol 1, pp 1291–1304 (1991)Google Scholar
  28. Schaub, H.: Relative orbit geometry through classical orbit element differences. J Guid. Control Dyn. 27(5), 839–848 (2004)CrossRefGoogle Scholar
  29. Schaub, H., Alfriend, K.T.: \(J_{2}\) invariant relative orbits for spacecraft formations. Celest. Mech. Dyn. Astron. 79(2), 77–95 (2001)ADSzbMATHCrossRefGoogle Scholar
  30. Schaub, H., Junkins, J.L.: Analytical Mechanics of Space Systems, 2nd edn. AIAA Education Series, Reston, VA (2009)Google Scholar
  31. Shapiro, B.E.: Phase plane analysis and observed frozen orbit for the Topex/Poseidon mission. Adv. Astronaut. Sci. 91, 853–872 (1995)Google Scholar
  32. Vadali, S.R.: An analytical solution for relative motion of satellites. In: 5th Dynamics and Control of Systems and Structures in Space Conference, Cranfield University, Cranfield, UK (2002)Google Scholar
  33. Vadali, S.R., Schaub, H., Alfriend, K.T.: Initial conditions and fuel-optimal control for formation flying of satellites. AIAA GNC Conference, Portland, Oregon AIAA Paper 99–4265 (1999)Google Scholar
  34. Xu, M., Wang, Y., Xu, S.: On the existence of \(J_2\) invariant relative orbits from the dynamical system point of view. Celest. Mech. Dyn. Astron. 112, 427–444 (2012)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Distributed Space Systems Lab, Faculty of Aerospace EngineeringTechnion-Israel Institute of TechnologyHaifaIsrael
  2. 2.Columnas de Hercules 1San FernandoSpain

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