The Lagrange coefficients of Vinti theory


The existence of Lagrange coefficients under two-body dynamics is well known, and the concept forms the basis of robust algorithms for solving a variety of fundamental astrodynamics problems. The Lagrange coefficients are generalized to Vinti theory in the present work, where the Vinti potential describes the dynamics of small objects like spacecraft orbiting an oblate body. Exact expressions for the coefficients are provided in various forms alongside detailed derivations. Useful properties and geometrical interpretations are also established.

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

Change history

  • 25 June 2020

    In the first paragraph of the Introduction, the last sentence should read as “ Vinti.


  1. Bate, R.R., Mueller, D.D., White, J.E.: Fundamentals of Astrodynamics. Dover Publications Inc, Mineola (1971)

    Google Scholar 

  2. Battin, R.H.: An Introduction to the Mathematics and Methods of Astrodynamics. AIAA Education Series, revised edn. American Institute of Aeronautics and Astronautics Inc., Reston (1999)

    Google Scholar 

  3. Biria, A.D.: Revisiting Vinti theory: generalized equinoctial elements and applications to spacecraft relative motion. Ph.D. thesis, Supervisor: Dr. Ryan P. Russell, The Univrsity of Texas at Austin, Austin, TX (2017)

  4. Biria, A.D., Russell, R.P.: A satellite relative motion model including \(J_2\) and \(J_3\) via Vinti’s intermediary. In: AAS/AIAA Space Flight Mechanics Meeting, Univelt, Inc., San Diego, CA, Advances in the Astronautical Sciences, vol. 158, pp. 3475–3494 (2016). (Paper AAS 16-537)

  5. Biria, A.D., Russell, R.P.: Equinoctial elements for Vinti theory: generalizations to an oblate spheroidal geometry. Acta Astronaut. 153, 274–288 (2018a).

    ADS  Article  Google Scholar 

  6. Biria, A.D., Russell, R.P.: A satellite relative motion model including \(J_2\) and \(J_3\) via Vinti’s intermediary. Celest. Mech. Dyn. Astron. (2018b).

    Article  MATH  Google Scholar 

  7. Biria, A.D., Russell, R.P.: Analytical solution to the Vinti problem in oblate spheroidal equinoctial orbital elements. J. Astronaut. Sci. (2019).

    Article  Google Scholar 

  8. Bonavito, N.L.: Computational Procedure for Vinti’s Theory of an Accurate Intermediary Orbit. Technical report TN D-1177, National Aeronautics and Space Administration, Washington, D.C. (1962)

  9. Der, G.J., Bonavito, N.L. (eds.): Orbital and Celestial Mechanics, Progress in Astronautics and Aeronautics, vol. 177. American Institute of Aeronautics and Astronautics, Reston (1998)

    Google Scholar 

  10. Getchell, B.C.: Orbit computation with the Vinti potential and universal variables. J. Spacecr. Rockets 7(4), 405–408 (1970).

    ADS  Article  Google Scholar 

  11. Herrick, S.: Universal variables. Astron. J. 70(4), 309–315 (1965).

    ADS  MathSciNet  Article  Google Scholar 

  12. Pitkin, E.T.: Integration and Optimization of Sustained-thrust Rocket Orbits. Ph.D. thesis, University of California, Los Angeles (1964)

  13. Pitkin, E.T.: A regularized approach to universal orbit variables. AIAA J. 3(8), 1508–1511 (1965).

    ADS  Article  Google Scholar 

  14. Stumpff, K.: Neue formeln und hilfstafeln zur ephemeridenrechnung. Astron. Nachr. 275(7), 108–128 (1947).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  15. Sundman, K.F.: Mémoire sur le problème des trois corps. Acta Math. 36, 105–179 (1913).

    MathSciNet  Article  MATH  Google Scholar 

  16. Vinti, J.P.: New method of solution for unretarded satellite orbits. J. Res. Natl. Bur. Stand. 63B(2), 105–116 (1959).

    MathSciNet  Article  Google Scholar 

  17. Vinti, J.P.: Theory of an accurate intermediary orbit for satellite astronomy. J. Res. Natl. Bur. Stand. 65B(3), 169–201 (1961).

    MathSciNet  Article  MATH  Google Scholar 

  18. Vinti, J.P.: Inclusion of the third zonal harmonic in an accurate reference orbit of an artificial satellite. J. Res. Natl. Bur. Stand. 70B(1), 17–46 (1966a).

    MathSciNet  Article  MATH  Google Scholar 

  19. Vinti, J.P.: Invariant properties of the spheroidal potential of an oblate planet. J. Res. Natl. Bur. Stand. 70B(1), 1–16 (1966b).

    MathSciNet  Article  MATH  Google Scholar 

  20. Vinti, J.P.: Improvement of the spheroidal method for artificial satellites. Astron. J. 74(1), 25–34 (1969).

    ADS  Article  MATH  Google Scholar 

  21. Wiesel, W.E.: Numerical solution to Vinti’s problem. J. Guid. Control Dyn. 38(9), 1757–1764 (2015).

    ADS  Article  Google Scholar 

  22. Wright, S.P.: Orbit determination using Vinti’s solution. Ph.D. thesis, Supervisor: Dr. William E. Wiesel, Air Force Institute of Technology, Wright-Patterson Air Force Base, OH (2016)

  23. Zurita, L.D.: Orbital resonances in the Vinti solution. In: Advanced Maui Optical and Space Surveillance Technologies (AMOS) Conference, Maui, Hawaii (2017)

Download references

Author information



Corresponding author

Correspondence to Ashley D. Biria.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The original version of this article was revised: the detailed corrections have been given in the correction article.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Biria, A.D. The Lagrange coefficients of Vinti theory. Celest Mech Dyn Astr 132, 26 (2020).

Download citation


  • Lagrange coefficients
  • Vinti theory
  • Spheroidal method
  • Intermediary