A lower bound of the distance between two elliptic orbits

  • Denis V. MikryukovEmail author
  • Roman V. Baluev
Original Article


We obtain a lower bound of the distance function (MOID) between two noncoplanar bounded Keplerian orbits (either circular or elliptic) with a common focus. This lower bound is positive and vanishes if and only if the orbits intersect. It is expressed explicitly, using only elementary functions of orbital elements, and allows us to significantly increase the speed of processing for large asteroid catalogs. Benchmarks confirm high practical benefits of the lower bound constructed.


Elliptic orbits MOID Linking coefficient Distance function Catalogs Asteroids and comets Near-Earth asteroids Space debris Close encounters Collisions 



We are grateful to Professor K. V. Kholshevnikov for the statement of the problem, for important remarks and for his help in preparing the manuscript. We also express our gratitude to A. Ravsky for valuable discussion, as well as anonymous reviewers, whose constructive and valuable comments greatly helped the authors to improve the manuscript. All calculations made in the work were conducted by means of the equipment of the Computing Centre of Research Park of Saint Petersburg State University. This work is supported by the Russian Science Foundation Grant No. 18-12-00050.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflicts of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed consent

This research did not involve human participants.


  1. Armellin, R., Di Lizia, P., Berz, M., Makino, K.: Computing the critical points of the distance function between two Keplerian orbits via rigorous global optimization. Celest. Mech. Dyn. Astron. 107, 377–395 (2010)ADSMathSciNetCrossRefGoogle Scholar
  2. Baluev, R.V., Mikryukov, D.V.: Fast error-controlling MOID computation for confocal elliptic orbits. Astron. Comput. 27, 11–22 (2019)ADSCrossRefGoogle Scholar
  3. Crowell, R.H., Fox, R.H.: Introduction to Knot Theory. Springer, Berlin (1963)zbMATHGoogle Scholar
  4. Dybczyński, P.A., Jopek, T.J., Serafin, R.A.: On the minimum distance between two Keplerian orbits with a common focus. Celest. Mech. 38, 345–356 (1986)ADSCrossRefGoogle Scholar
  5. Gellert, W., Gottwald, S., Hellwich, M., Kästner, H., Küstner, H.: The VNR Concise Encyclopedia of Mathematics, 2nd edn. Van Nostrand Reinhold, New York (1989)Google Scholar
  6. Gronchi, G.F.: On the stationary points of the squared distance between two ellipses with a common focus. SIAM J. Sci. Comput. 24(1), 61–80 (2002)MathSciNetCrossRefGoogle Scholar
  7. Gronchi, G.F.: An algebraic method to compute the critical points of the distance function between two Keplerian orbits. Celest. Mech. Dyn. Astron. 93, 295–329 (2005)ADSMathSciNetCrossRefGoogle Scholar
  8. Hedo, J.M., Ruíz, M., Peláez, J.: On the minimum orbital intersection distance computation: a new effective method. Mon. Not. R. Astron. Soc. 479(3), 3288–3299 (2018)ADSCrossRefGoogle Scholar
  9. Kholshevnikov, K.V., Titov, V.B.: Two-Body Problem: The Tutorial. St. Petersburg State University Press, St. Petersburg (2007) (in Russian)Google Scholar
  10. Kholshevnikov, K.V., Vassiliev, N.N.: On linking coefficient of two Keplerian orbits. Celest. Mech. Dyn. Astron. 75, 67–74 (1999a)ADSMathSciNetCrossRefGoogle Scholar
  11. Kholshevnikov, K.V., Vassiliev, N.N.: On the distance function between two Keplerian elliptic orbits. Celest. Mech. Dyn. Astron. 75, 75–83 (1999b)ADSMathSciNetCrossRefGoogle Scholar
  12. Sitarski, G.: Approaches of the parabolic comets to the outer planets. Acta Astron. 18(2), 171–195 (1968)ADSGoogle Scholar
  13. Vassiliev, N.N.: Determining of critical points of distance function between points of two Keplerian orbits. Bull. Inst. Theor. Astron. 14(5), 266–268 (1978)ADSGoogle Scholar
  14. Zheleznov, N.B., Kochetova, O.M., Kuznetsov, V.B., Medvedev, Yu.D., Chernetenko, Yu.A., Shor, V.A.: Ephemerides of minor planets for 2018. St. Petersburg, Inst. Appl. Astron. (2017)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Central Astronomical Observatory at Pulkovo of the Russian Academy of SciencesSt. PetersburgRussia

Personalised recommendations