Skip to main content
SpringerLink
Log in

Generalized Problem of Two Fixed Centers or the Darboux-Gredeaks Problem

  • Published:
Cosmic Research Aims and scope Submit manuscript

Abstract

A brief review of publications on the problem of two fixed centers is given, including its generalizations and astronomical applications. A comparison of the Darboux potential with that of Eve Gredeaks is made. An account of the basic points of development of modern high-precision theories of the motion of planetary satellites, based on the Gredeaks’ intermediate orbit, is given.

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

Access this article

Log in via an institution

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

  1. Aksenov, E.P., Grebenikov, E.A., and Demin, V.G., General Solution to the Problem of Satellite Motion in the Normal Field of Earth’s Attraction, Iskusstv. Sputniki Zemli, 1961, no. 8, p. 64.

  2. Aksenov, E.P., Asymmetrical Intermediate Orbits of Earth’s Satellites, Soobshcheniya Gos. Astron. Inst. im. P.K. Shternberga, 1968, no. 155, pp. 3–51.

  3. Aksenov, E.P., Teoriya dvizheniya iskusstvennykh sputnikov Zemli (The Theory of Motion of Artificial Earth’s Satellites), Moscow: Nauka, 1977.

    Google Scholar 

  4. Aksenov, E.P., Vashkov’yak, S.N., and Emel’yanov, N.V., Orbit Elements of Satellites of the ISAZhEKS Program, Nablyudeniya ISZ, 1978, no. 18, pp. 277–294.

  5. Alekseev, V.M., Generalized Three-Dimensions Problem of Two Fixed Centers: Classification of Motions, Byull. Inst. Teor. Astron., 1965, vol. 10, no.4, p. 117.

    Google Scholar 

  6. Arazov, G.T., Investigation of Motion of a Satellite of Spheroidal Planet, Astron. Zh., 1975, vol. 52, no.4, pp. 891–894.

    Google Scholar 

  7. Badalyan, G.K., Problem of Two Fixed Centers, Astron. Zh., 1934, vol. 11, no.4, pp. 346–378.

    Google Scholar 

  8. Burshtein, E.I., Solutions to the Laplace Equation in Ellipsoidal Coordinates and Their Degeneracies Satisfying the Conditions of the Steckel Theorem, Available from VINITI, 1972, no. 4577.

  9. Emel’yanov, N.V., Perturbations of the 3rd and 4th Order Relative to Planet’s Oblateness in the Satellite Orbit, Astron. Zh., 1979, vol. 56, no.5, pp. 1070–1076.

    Google Scholar 

  10. Emel’yanov, N.V. and Nasonova, L.P., Expansion of Perturbing Function Caused by the Earth’s Nonsphericity, Astron. Zh., 1984, vol. 61, no.5, pp. 1021–1028.

    Google Scholar 

  11. Emel’yanov, N.V., Construction of Analytical Theory of Satellite Motion with an Accuracy to the Third Order Relative to the Earth’s Oblateness, Astron. Zh., 1986, vol. 63, no.4, pp. 800–809.

    Google Scholar 

  12. Emel’yanov, N.V., Vashkov’yak, S.N., and Nasonova, L.P., Results of Determination from Observations of Dynamical and Kinematical Parameters of the System of Martian Satellites, Astron. Zh., 1992, vol. 659, no.4, pp. 863–871.

    Google Scholar 

  13. Demin, V.G., Dvizhenie iskusstvennogo sputnika v netsentral’nom pole tyagoteniya (Motion of Artificial Satellite in a Noncentral Gravitational Field), Moscow: Nauka, 1968.

    Google Scholar 

  14. Kaisin, V.K., Spacecraft Motion in the Normal Gravity Field of the Earth under the Action of Additional Forces, Kosm. Issled., 1969, vol. 7, no.5, pp. 686–693.

    Google Scholar 

  15. Kaisin, V.K., One Case of Generalization of the Problem of Two Immobile Centers, Byull. Inst. Teor. Astron., 1970, vol. 12, no.2, p. 163.

    Google Scholar 

  16. Kaisin, V.K., Qualitative Analysis of the Forms of Motion of a Star in a Stationary Star System with an Axisymmetric Nucleus, I, Astron. Zh., 1975, vol. 52, no.4, pp. 878–882.

    Google Scholar 

  17. Kozlov, I.S., Interpretation and Applications of the Problem of Four Fixed Centers, Astron. Zh., 1975, vol. 52, no.3, pp. 649–656.

    Google Scholar 

  18. Koman, G.G., Intermediate Orbits of Artificial Lunar Satellites, Soobshcheniya Gos. Astron. Inst. im. P.K. Shternberga, 1973, no. 186, p. 3.

  19. Koman, G.G., One Form of Differential Equations of Motion of Artificial Lunar Satellites, Astron. Zh., 1975, vol. 52, no.1, pp. 207–209.

    Google Scholar 

  20. Kunitsyn, A.L., Qualitative Investigation of Motion in One Limiting Variant of the Problem of Two Fixed Centers, Trudy Univ. Druzhby Narodov, Ser. Teor. Mekh., 1966, vol. 17, no.4, p. 32.

    Google Scholar 

  21. Tamarov, V.A., Expansion of Perturbing Function in the Problem of Two Fixed Centers, Astron. Zh., 1982, vol. 59, pp. 779–787.

    Google Scholar 

  22. Tatarinov, Ya.V., Lektsii po klassicheskoi dinamike (Lectures in Classical Dynamics), Moscow: Izd. Mosk. Univ., 1984.

    Google Scholar 

  23. Timoshkova, E.I., Expansion of Perturbing Function for the Case of Noncentral Body, Vestn. Leningr. Univ., 1972a, no. 49, pp. 137–142.

  24. Timoshkova, E.I., Concerning the Problem of Expansion of perturbation Function, Astron. Zh., 1972b, vol. 49, pp. 879–885.

    Google Scholar 

  25. Chazov, V.V., Calculation of Motion of Artificial Satellites in the Earth’s Gravity Field, Nauchnye Informatsii Astronomicheskogo Soveta AN SSSR, 1987, vol. 62, pp. 127–132.

    Google Scholar 

  26. Chazov, V.V., One Possibility of the Use of Intermediate Orbit in Perturbation Theory, Trudy Gos. Astron. Inst. im. P.K. Shternberga, 1988, vol. 60, pp. 7–15.

    Google Scholar 

  27. Chazov, V.V., Basic Algorithms of Numerical-Analytical Theory of Motion of Earth’s Artificial Satellites, Trudy Gos. Astron. Inst. im. P.K. Shternberga, 2000, vol. 68, pp. 5–20.

    Google Scholar 

  28. Jakobi, K., Lektsii po dinamike (Lectures in Dynamics), Moscow: ONTI, 1936.

    Google Scholar 

  29. Burrau, C., Uber Einige in Aussicht Genommene Berechnung, Betreffend einen Spezialfall des Dreikorperproblems, Vierteljahrschrift Astron. Ges., 1906, vol. 41, p. 261.

    Google Scholar 

  30. Cook, A.H., The Motion of Artificial Satellites in the Gravitational Field of Earth, Bologna, 1967, p. 96.

  31. Darboux, G., Sur un Probleme de Mechanique, Archives Neerlandaises des Sciences, Exact et Naturel, 1901, ser. 2, vol. 6, p. 371.

    Google Scholar 

  32. Euler, L., Un Corps Etant Attire an Raison Reciproque Quarree des Distances Vers Deux Points Fixes Donnes, Mem. Berlin, 1760, vol. 228.

  33. Euler, L., De Motu Corporis ad Duo Centra Virium Fixa Attracti, Novi Commet. Acad. Scient. Imperial. Petropolit., 1764, vol. 10, p. 207; 1765, vol. 11, p. 152.

    Google Scholar 

  34. Hiltebeitel, A.M., On the Problem of Two Fixed Centers and Certain of Its Generalizations, Am. J. Math., 1911, vol. 33, p. 337.

    Google Scholar 

  35. Introduction aux Ephemerides Astronomiques. Bureau des Longitudes, Paris: Les Editions de Physique, 1997.

  36. Kozai, Y., Spec. Rept. Smithsonian Astrophys. Observ., 1964, no. 165.

  37. Lagrange, J., Recherhes sur le Mouvement d’un Corp’s qui Attire par deux Centres Fixes, Oeuvres de Lagrange, vol. 2, p. 67. Or Mem. De Turin, 1766–1769, vol. 4, pp. 118, 215.

    Google Scholar 

  38. Liouville, J., Sur Quelques Cas Particuliers ou les Equations du Mouvement d’Un Point Materiel Peuvent S’Integrer, J. Math., 1846, vol. 11, p. 345.

    Google Scholar 

  39. Newton, R.R., Motion of a Satellite around an Unsymmetrical Central Body, J. Appl. Phys., 1959, vol. 30, no.2, p. 115.

    Google Scholar 

  40. Tallqvist, H., Zum Zweizentrenproblems, Acta Societatis Scientiarum Fennicae. Nova Series A, 1927, vol. 1, no.1.

  41. Thiele, T.N., Recherhes Numeriques Consernant des Solutions Periodiques d’Un Cas Special du Probleme des Trois Corps, Astron. Nachrichten, 1896, vol. 138, p. 1.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sternberg State Astronomical Institute, Universitetskii pr. 13, Moscow, 119899, Russia

    L. G. Lukyanov, N. V. Emeljanov & G. I. Shirmin

Authors
  1. L. G. Lukyanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. V. Emeljanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. I. Shirmin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to the memory of Professor Evgenii Petrovich Aksenov

__________

Translated from Kosmicheskie Issledovaniya, Vol. 43, No. 3, 2005, pp. 194–200.

Original Russian Text Copyright © 2005 by Lukyanov, Emeljanov, Shirmin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lukyanov, L.G., Emeljanov, N.V. & Shirmin, G.I. Generalized Problem of Two Fixed Centers or the Darboux-Gredeaks Problem. Cosmic Res 43, 186–191 (2005). https://doi.org/10.1007/s10604-005-0033-5

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10604-005-0033-5

Keywords

Search

Navigation