Abstract
When the perturbation affecting a Keplerian motion is proportional tor −n (n≥3), a canonical transformation of Lie type will convert the system into one in which the perturbation is proportional tor −2. Because it removes parallactic factors, the transformation is called the elimination of the parallax.
In the main problem for the theory of artificial satellites, the elimination of the parallax has been conducted by computer to order 4. The first order in the reduced system may now be integrated in closed form, thereby revealing the fundamental property of the first-order intermediary orbits in line with Newton's Propositio XLIV.
Extension beyond order 1 leads to identify a new class of intermediaries for the main problem in nodal coordinates, namely the radial intermediaries.
The technique of smoothing a perturbation prior to normalizing the perturbed Keplerian system, of which the elimination of the parallax is an instance, is applied to derive the intermediaries in nodal coordinates proposed by Sterne, Garfinkel, Cid-Palacios and Aksnes, and to find the canonical diffeomorphisms which relate them to one another and to the radial intermediaries.
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References
Abraham, R. and Marsden, J. E.: 1978,Foundations of Mechanics, 2nd edition, Benjamin Cummings Publishing Co., London-New York.
Aksnes, K.: 1965,Astroph. Norv. 10, 69.
Aksnes, K.: 1966,Astroph. Norv. 10, 149.
Aksnes, K.: 1969, Ph.D. dissertation, Graduate School of Yale University, New Haven, Connecticut.
Aksnes, K.: 1970,Astron. J. 75, 1066.
Andoyer, H.: 1913,Bull. Astron. 30, 28.
Arnold, V. I.: 1978,Mathematical Methods of Classical Mechanics, translated from Russian by K. Vogtmann and A. Weinstein, Springer-Verlag, New York.
Brouwer, D.: 1959,Astron. J. 64, 378.
Brouwer, D. and Clemence, G. M.: 1961,Methods of Celestial Mechanics, Academic press, New York and London.
Cefola, P.: 1974, AIAA 12th Aerospace Sciences Meeting, Washington D. C., Paper No 74-170.
Chazy, J.: 1953,Mécanique céleste-Equations canoniques et variations des constantes. Presses universitaires de France, Paris.
Cid-Palacios, R.: 1969,Publ. Rev. Ac. Cienc. Zaragoza 24, 159.
Claus, A. J. and Lubowe, A. G.: 1963,Astronaut Acta 9, 275.
Coffey, S. L.: 1980, Manuscript in private communication prior to publication.
Dasenbrock, R.: 1973, NRL Report 7564, Naval Research Laboratory, Washington D. C. 20375.
Deprit, A.: 1969,Celes. Mech. 1, 12.
Deprit, A.: 1975,J. Res. Natl. Bur. Std. 79B, 1.
Deprit, A.: 1978, AIAA Paper 78-1439, AIAA/AAS Astrodynamics Conference, August 7–9, Palo Alto, California. Accepted for publication in theJourn. for Guidance and Control.
Deprit, A.: 1979,Celes. Mech. 20, 325.
Deprit, A.: 1981,Celes. Mech. 24, 185 (this issue).
Deprit, A. and Richardson, D.: 1981,Celes. Mech. (submitted).
Dieudonné, J.: 1960,Foundations of Modern Analysis, Academic Press, New York
Garfinkel, B.: 1958,Astron. J. 63, 88.
Garfinkel, B.: 1959,Astron. J. 64, 353.
Garfinkel, B.: 1964,Astron. J. 69, 223.
Garfinkel, B. and Aksnes, K.: 1970,Astron. J. 75, 85.
Goscinny, R. and Uderzo, A. 1972:Le Devin, Dargaud éditeur, Paris.
Henrard, J.: 1970,Celes. Mech. 3, 107.
Hill, G. W.: 1902,Astron. J. 22, 183.
Hori, G.: 1966,Publ. Astron. Soc. Japan 18, 287.
Kernighan, B. W. and Plauger, P. J.: 1976,Software Tools, Addison-Wesley Publishing Co., Reading.
Kislik, M. D.: 1961,Artificial Earth Satellites 4, 183.
Koyré, A. and Cohen, I. B.: 1972,Isaac Newton's Philosophiae Naturalis Principia Mathematica, Harvard University Press.
Kozai, Y.: 1962,Astron. J. 67, 446.
Lubowe, A. G.: 1964,Astron. Acta 10, 253.
Lubowe, A. G.: 1966,AIAA J. 4, 361.
Poincaré, H.: 1889,Comptes rendus Acad. Sc. Paris,108, 210; also inOeuvres de Henri Poincaré, t. VII, 551, Gauthier-Villars, Paris.
Poincaré, H.: 1891,Comptes rendus Acad. Sc. Paris,114, 1305; also inOeuvres de Henri Poincaré, t. VII 491, Gauthier-Villars, Paris.
Poincaré, H.: 1893,Les méthodes nouvelles de la mécanique céleste, t. II, pp. 17–24, Gauthier-Villars, Paris.
Poincaré, H.: 1897,Bull. Astron. 14, 241; also inOeuvres de Henri Poincaré, t. VII, 516, Gauthier-Villars, Paris.
Rom, A.: 1969,Celes. Mech. 1, 301.
Shniad, H.: 1970,Celes. Mech. 2, 114.
Smart, W. M.: 1953,Celestial Mechanics, Longmans, Green and Co., London.
Sterne, T. E.: 1957,Astron. J. 63, 28.
Sterne, T. E.: 1960,An Introduction to Celestial Mechanics, Interscience Publishers Ltd., London.
Tisserand, P.: 1894,Traité de mécanique céleste, t. III, Gauthier-Villars, Paris.
Vinti, J. P.: 1959,J. Res. Nat. Bur. Standards 63B, 105.
Whittaker, E. T.: 1904,A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press.
Wu Lian-da, Wang Chang-bin and Tong-Fu: 1978,Acta Astron. Sinica 19, 131; English translation inChinese Astron. 3, 238.
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Deprit, A. The elimination of the parallax in satellite theory. Celestial Mechanics 24, 111–153 (1981). https://doi.org/10.1007/BF01229192
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DOI: https://doi.org/10.1007/BF01229192