Skip to main content
Log in

Solutions and periodicity of satellite relative motion under even zonal harmonics perturbations

  • Original Article
  • Published:
Celestial Mechanics and Dynamical Astronomy Aims and scope Submit manuscript

Abstract

Finding relative satellite orbits that guarantee long-term bounded relative motion is important for cluster flight, wherein a group of satellites remain within bounded distances while applying very few formationkeeping maneuvers. However, most existing astrodynamical approaches utilize mean orbital elements for detecting bounded relative orbits, and therefore cannot guarantee long-term boundedness under realistic gravitational models. The main purpose of the present paper is to develop analytical methods for designing long-term bounded relative orbits under high-order gravitational perturbations. The key underlying observation is that in the presence of arbitrarily high-order even zonal harmonics perturbations, the dynamics are superintegrable for equatorial orbits. When only J 2 is considered, the current paper offers a closed-form solution for the relative motion in the equatorial plane using elliptic integrals. Moreover, necessary and sufficient periodicity conditions for the relative motion are determined. The proposed methodology for the J 2-perturbed relative motion is then extended to non-equatorial orbits and to the case of any high-order even zonal harmonics (J 2n , n ≥ 1). Numerical simulations show how the suggested methodology can be implemented for designing bounded relative quasiperiodic orbits in the presence of the complete zonal part of the gravitational potential.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abramowitz M., Stegun I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1964)

    MATH  Google Scholar 

  • Alfriend K.T., Vadali S.R., GurfilP. How J.P., Breger L.: Spacecraft Formation Flying: Dynamics, Control and Navigation, Chaps. I, IV, V, VI and VII. Elsevier, Oxford (2010)

    Google Scholar 

  • Arnold V.I., Kozlov V., Neishtadt A.I.: Mathematical Aspects of Classical and Celestial Mechanics. 3rd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  • Battin R.H.: An Introduction to the Mathematics and Methods of Astrodynamics. AIAA, Reston (1999)

    MATH  Google Scholar 

  • Ben-Ya’acov, U.: Laplace-Runge-Lenz symmetry in general rotationally symmetric systems. ArXiv e-prints (2010)

  • Brouwer D.: Solution of the problem of artificial satellite theory without drag. Astron. J. 64, 378–397 (1959)

    Article  MathSciNet  ADS  Google Scholar 

  • Brouwer D., Clemence G.M.: Methods of Celestial Mechanics. Academic Press, New York (1961)

    Google Scholar 

  • Brown, O., Eremenko, P.: Fractionated space architectures: a vision for responsive space. In: 4th Responsive Space Conference, Los Angeles, CA, (2006)

  • Calkin M.G.: Lagrangian and Hamiltonian mechanics. World Scientific, Singapore (1996)

    MATH  Google Scholar 

  • Celletti A., Negrini P.: Non-integrability of the problem of motion around an oblate planet. Celest Mech Dyn Astron 61, 253–260 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Condurache, D.: New symbolic procedures in the study of the dynamical systems. Ph.D. Thesis, Technical University “Gheorghe Asachi”, Iasi, Romania (1995)

  • Condurache D., Martinuşi V.: Relative spacecraft motion in a central force field. J. Guid. Control Dyn. 30(3), 873–876 (2007)

    Article  Google Scholar 

  • Condurache, D., Martinuşi, V.: Exact solution to the relative orbital motion in a central force field. In: 2nd International Symposium on Systems and Control in Aerospace and Astronautics, Shenzen, China, pp. 1–6. IEEE, China (2008)

  • Condurache D., Martinuşi V.: Foucault pendulum-like problems: a tensorial approach. Int. J. Non-linear Mech. 43(8), 743–760 (2008)

    Article  ADS  MATH  Google Scholar 

  • Condurache, D., Martinuşi, V.: Super-integrability in the unperturbed relative orbital motion problem. In: AIAA Guidance, Navigation, and Control Conference, Toronto, Canada, paper AIAA 2010-7669 (2010)

  • 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)

    Article  Google Scholar 

  • Gim D.-W., Alfriend K.T.: Satellite relative motion using differential equinoctial elements. Celest. Mech. Dyn. Astron. 92(4), 295–336 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Gurfil P.: Relative motion between elliptic orbits: generalized boundedness conditions and optimal formationkeeping. J. Guid. Control Dyn. 28(4), 761–767 (2005)

    Article  MathSciNet  Google Scholar 

  • Gurfil P., Kholshevnikov K.V.: Manifolds and metrics in the relative spacecraft motion problem. J. Guid. Control Dyn. 29(4), 1004–1010 (2006)

    Article  Google Scholar 

  • Hamel J.F., de Lafontaine J.: Linearized dynamics of formation flying spacecraft on a J 2-perturbed elliptical orbit. J. Guid. Control Dyn. 30(6), 1649–1658 (2007)

    Article  Google Scholar 

  • Irigoyen M., Simo C.: Nonintegrability of the J 2 problem. Celest. Mech. Dyn. Astron. 55, 281–287 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Jezewski D.J.: An analytic solution for the J 2 perturbed equatorial orbit. Celest. Mech. 30, 363–371 (1983)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Leach P.G.L., Flessas G.P.: Generalisations of the Laplace-Runge-Lenz vector. J. Nonlinear Math. Phys. 10(3), 340–423 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Ross I.M.: Linearized dynamic equations for spacecraft subject to J 2 perturbations. J. Guid. Control Dyn. 26(4), 657–659 (2003)

    Article  Google Scholar 

  • Schaub H., Alfriend K.T.: J 2 invariant relative orbits for spacecraft formations. Celest. Mech. Dyn. Astron. 79(2), 77–95 (2001)

    Article  ADS  MATH  Google Scholar 

  • Schaub H., Junkins J.L.: Analytical Mechanics of Space Systems. 2nd edn. AIAA Education Series, Reston (2009)

    MATH  Google Scholar 

  • Schweighart S.A., Sedwick R.J.: High-fidelity linearized J 2 model for satellite formation flight. J. Guid. Control Dyn. 25(6), 1073–1080 (2002)

    Article  Google Scholar 

  • Sengupta P., Vadali S.R., Alfriend K.T.: Second-order state transition for relative motion near perturbed, elliptic orbits. Celest. Mech. Dyn. Astron. 97(2), 101–129 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Uldall Kristiansen K., Palmer P.L., Roberts M.: Relative motion of satellites exploiting the super-integrability of Kepler’s problem. Celest. Mech. Dyn. Astron. 106(4), 371–390 (2009)

    Article  Google Scholar 

  • Vadali, S.R.: An analytical solution for relative motion of satellites. In: 5th Dynamics and Control of Systems and Structures in Space Conference, Cranfield, UK, Cranfield University (2002)

  • Vallado D.A.: Fundamentals of Astrodynamics and Applications. 2nd edn. Microcosm, USA (2001)

    Google Scholar 

  • Wiesel W.E.: Relative satellite motion about an oblate planet. J. Guid. Control Dyn. 25(4), 776–785 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Pini Gurfil.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Martinuşi, V., Gurfil, P. Solutions and periodicity of satellite relative motion under even zonal harmonics perturbations. Celest Mech Dyn Astr 111, 387–414 (2011). https://doi.org/10.1007/s10569-011-9376-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10569-011-9376-9

Keywords

Navigation