The Nature of the Critical Inclinations in Artificial Satellite Theory

  • Shannon L. Coffey
  • Andre Deprit
  • Bruce R. Miller
Part of the Astrophysics and Space Science Library book series (ASSL, volume 127)


Mainfolds of orbits with stationary perigees are intrinsic features of the averaged main problem in aritifical satellite theory: they bifurcate off the manifold of circular orbits at the points where stability flips to instability and vice-versa.


Circular Orbit North Pole Meridian Plane Keplerian Problem Reduce Phase Space 
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  1. 1a.
    Krause, H.G.L. “Die Sakularstorungen ciner Aussenstationsbahn” in Probleme aus der Astronnautischen Grundlagenforschung. ed. H.H.Kolle, Vortrage III Internationalen Astronautischen Kongress. Stuttgart 1–6 September 1952, 161–173;Google Scholar
  2. 1b.
    Roberson, R.E., J. Franklin Inst. 264, 161–202 and 269–285(1957);Google Scholar
  3. 1c.
    Petty, C.M., and Breakwell, J.V., J. Franklin Inst. 270, 259–282(1960).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 2.
    Orlov, A.A., ?? 3,3–38(T953). ??Google Scholar
  5. 3a.
    Brouwer, D., Astron. J. 51, 223–231(1946),MathSciNetADSCrossRefGoogle Scholar
  6. 3b.
    63, 133–138(1958);MathSciNetADSGoogle Scholar
  7. 3c.
    Herget, P. and Musen, P., Astron. J. 63, 430–433(1958);ADSCrossRefGoogle Scholar
  8. 3d.
    Brouwer,D., 64, 378–397(1959).MathSciNetADSCrossRefGoogle Scholar
  9. 4a.
    Geyling, F.T., Astronautica Acta 11, 196–201 (1965);zbMATHGoogle Scholar
  10. 4b.
    Lubowe, A.G., Celest. Mech. 1, 6–20 and 143(1969).ADSCrossRefGoogle Scholar
  11. 5a.
    Message, P.J., Hori,G., and Garfinkel, B., The Observatory 82,168–170 (1962);ADSGoogle Scholar
  12. 5b.
    Kikuchi, S., Astronom. Nachr. 289, 241–245(1967);ADSzbMATHGoogle Scholar
  13. 5c.
    Garfinkel, B., Celest. Mech. 1,11(1969);ADSCrossRefGoogle Scholar
  14. 5d.
    Hughes,S., Celest. Mech. 25, 235–266 (1981).ADSzbMATHCrossRefGoogle Scholar
  15. 6.
    First proposed by K.R. Meyer in Dynamical Systems, ed. M.M. Peixoto, Academic Press, New York, 259–272 (1973), and a year later by J. Marsden and A. Weinstein in Rep. Math. Phys. 5, 121–130(1974).The equivalent of Meyer’s reduction in quantum physics being the quantization procedure of Kirillov, Souriau, and Kostant, text-books in physics have taken to designate the classical construction as the KSK-reduction.Google Scholar
  16. 7.
    As expressed in A., Deprit, Celest. Mech., 26, 9–21 (1982), the concept of a Delaunay normalization got justified in intrinsic terms by R. Cushman in Differential Geometric Methods in Mathematical Physics, ed. S. Sternberg, D. Reidel Publishing Company, Dordrecht, 125–144(1984).Google Scholar
  17. 8a.
    Deprit, A., Celest. Mech. 29, 229–247 (1983);MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. 8b.
    F.Mignard and M.Henon, Celes. Mech. 33, 239–250 (1984);MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 8c.
    A. Deprit in The Big Bang and Georges Lemaitre, ed. A. Berger, D. Reidel Publishing Company, Dordrecht, 151–180 (1984).Google Scholar
  20. 9.
    For another way of defining the projection P(L) → P(L,H) see R.Cushman, Celest. Mech.31, 401–429(1983).Google Scholar
  21. 10.
    These facts brought up first by H.G.L. Krause were later confirmed by R.E. Roberson, D. Brouwer, Astron. J. 63, 133–138(1958), and also, although somewhat inadvertently, by P. Herget and P. Musen, Astron.J. 63, 430–433 (1959).Google Scholar
  22. 11a.
    R.A. Strubble, Arch. Rational Mech. Anal. 7, 87–104 (1960);ADSCrossRefGoogle Scholar
  23. 11b.
    G.Hori, Astron. J. 65, 53 and 291–300 (1960).MathSciNetADSCrossRefGoogle Scholar
  24. 12.
    In regard to the origin and development of this LISP-based algebraic processor the reader is referred to a retrospective note by MACSYMA’s principal architect, J. Moses, in the Proceedings of the EUROSAM Conference ACM SIGSAM Bulletin, August 1974. A good introduction to the use of MACSYMA in nonlinear mechanics is to be found in R.H. Rand. Computer Algebra in Applied Mathematics. An Introduction to MACSYMA, Pitnam Advanced Publishing Program. Boston, MA (1984). As an illustration of its application to artificial satellite theory, see E. Zeis, A Computerized Algebraic Utility for the Construction of Non Singular Satellite Theory, M.S. Thesis Massachusetts Institute of Technology, Cambridge, MA (1978). Part of the thesis is published in E.Zeis, and P. Cefola., J.Guidance and Control 3, 48–54 (1978).Google Scholar
  25. 13.
    Our results complete the analysis begun by S. Aoki, Astron. J.67, 571–572 (1962), 68, 271–272, 355–365, 365–381 (1963),and continued by A.Jupp. Celest. Mech. 11, 361–378(1975) and 21, 361–393(1980).Google Scholar

Copyright information

© D. Reidel Publishing Company 1986

Authors and Affiliations

  • Shannon L. Coffey
    • 1
  • Andre Deprit
    • 2
  • Bruce R. Miller
    • 2
  1. 1.Naval Research LaboratoryUSA
  2. 2.National Bureau of StandardsGaithersburgUSA

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