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Stability of resonance rotation of a satellite with respect to its center of mass in the orbit plane

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An Erratum to this article was published on 01 October 2006

Abstract

The stability of resonance oscillations and rotations of a satellite in the plane of its orbit in the case when the difference of the moments of inertia with respect to the principal axes lying in the orbit plane is small is determined at a given rotation number m by the sign of function Φm(e), introduced by F.L. Chernous’ko in 1963. In this paper, convenient analytical representations of functions Φm(e) are described in the form of integrals and series of Bessel functions regular at e → 1−. Values of Φm(1) are calculated in explicit form. A theorem about the double asymptotic form of functions Φm(e) at m → ∞ and e → 1− is proved by the saddlepoint method.

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References

  1. Abalakin, V.K., Aksenov, E.P., Grebennikov, E.A., and Ryabov, Yu.A., Spravochnoe rukovodstvo po nebesnoi mekhanike i astrodinamike (A Handbook on Celestial Mechanics and Astrodynamics), Moscow: Nauka, 1971.

    Google Scholar 

  2. Beletskii, V.V., Libration of a Satellite, in Iskusstvennye sputniki Zemli. Vyp. 3 (Earth’s Artificial Satellites, Issue 3), Moscow: Akad. Nauk SSSR, 1959, pp. 13–31.

    Google Scholar 

  3. Beletskii, V.V., Dvizhenie iskusstvennogo sputnika otnositel’no tsentra mass (Motion of an Artificial Satellite with Respect to the Center of Mass), Moscow: Nauka, 1965.

    Google Scholar 

  4. Beletskii, V.V. and Lavrovskii, E.K., Theory of Resonance Rotation of Mercury, Astron. Zh., 1975, vol. 52, no. 6, pp. 1299–1308.

    ADS  Google Scholar 

  5. Beletskii, V.V. and Khentov, A.A., Rezonansnye vrashcheniya nebesnykh tel, (Resonance Rotations of Celestial Bodies), N. Novgorod: Nizhegorodskii gumanitarnyi tsentr, 1995.

    Google Scholar 

  6. Brauer, D. and Clemens, J., Metody nebesnoi mekhaniki (Methods of Celestial Mechanics), Moscow: Mir, 1964.

    Google Scholar 

  7. Bruno, A.D., Vibrations of a Satellite in Elliptical Orbit, Preprint of Keldysh Inst. of Applied Math., USSR Acad. Sci., Moscow, 1976, no. 53.

  8. Bruno, A.D., Lokal’nyi metod nelineinogo analiza differentsial’nykh uravnenii (A Local Method of Nonlinear Analysis of Differential Equations), Moscow: Nauka, 1979.

    Google Scholar 

  9. Bruno, A.D., Families of Periodic Solutions to the Beletsky Equation, Kosm. Issled., 2002, vol. 40, no. 3, pp. 295–316.

    Google Scholar 

  10. Bruno, A.D. and Varin, V.P., The Limit Problems for the Equation of Oscillations of a Satellite, Celestial Mechanics and Dynamical Astronomy, 1997, vol. 67, pp. 1–40.

    Article  ADS  MathSciNet  Google Scholar 

  11. Bruno, A.D. and Varin, V.P., Classes of Families of Generalized Periodic Solutions to the Beletsky Equation, Celestial Mechanics and Dynamical Astronomy, 2004, vol. 88, pp. 325–341.

    Article  ADS  MathSciNet  Google Scholar 

  12. Bruno, A.D. and Petrovich, V.Yu., Calculation of Periodic Vibrations of a Satellite: Regular Case, Preprint of Keldysh Inst. of Applied Math., Russ. Acad. Sci., Moscow, 1993, no. 65.

  13. Vinx, N.X., Sur les Solutions Periodiques du Mouvement Plan de Libration des Satellites et des Planetes, Celestial Mechanics, 1973, vol. 8, no. 3, pp. 371–403.

    ADS  MathSciNet  Google Scholar 

  14. Goldriech, P. and Peale, S., The Dynamics of Planetary Rotations, Annual Rev. Astron. and Astrophys., 1968, vol. 6, pp. 287–320.

    ADS  Google Scholar 

  15. Lutze, F.H, Jr., and Abbit, M.W., Jr. Rotational Locks for Near-Symmetric Satellites, Celestial Mechanics, 1969, vol. 1, no. 1, pp. 31–35.

    Article  ADS  Google Scholar 

  16. Sadov, S.Yu., Analysis of a Function Determining the Stability of Rotation of Almost Symmetrical Satellite, Preprint of Keldysh Inst. of Applied Math., Russ. Acad. Sci., Moscow, 1994, no. 84.

  17. Sadov, S.Yu., Coefficients of Averaged Equation of Vibrations of a Satellite, Preprint of Keldysh Inst. of Applied Math., Russ. Acad. Sci., Moscow, 1995, no. 27.

  18. Sadov, S.Yu., Application of Method of Averaging in a Singular Case, in Sovremennye metody nelineinogo analiza. Tezisy dokladov (Modern Methods of Nonlinear Analysis. Book of Abstracts), Voronezh: VGU, 1995, pp. 79–80.

    Google Scholar 

  19. Sadov, S.Yu., Analysis of Nonlinear Oscillations in Two Problems of Mechanics Using Normal Form, Cand. Sc. (Fiz.-Mat.) Dissertation, Moscow: Keldysh Inst. of Applied Math., Russ. Acad. Sci., 1995.

    Google Scholar 

  20. Sadov, S.Yu., Calculation of Normal Form of the Equation of Oscillations of a Satellite in a Singular Case, in Komp’yuternye metody v nebesnoi mekhanike. Tezisy dokladov (Computer Methods in Celestial Mechanics. Abstracts), St. Petersburg: ITA RAN, 1995.

    Google Scholar 

  21. Sadov, S.Yu., Computation of the Normal Form in a Singular Case, 1st Int. IMACS Conference on Appl. of Computer Algebra (Albuquerque, NM, 1995) Electronic Proceedings, Wester, M., Steinberg, S., and Jahn, M., Eds., http://math.unm.edu/ACA/1995/Proceedings/

  22. Sadov, S.Yu., Normal Form of the Equation of Satellite Oscillations in a Singular Case, Mat. Zametki, 1995, vol. 58, no. 5, pp. 785–789.

    MATH  MathSciNet  Google Scholar 

  23. Sadov, S.Yu., Higher Approximations of the Method of Averaging for the Equation of Planar Oscillations of a Satellite, Preprint of Keldysh Inst. of Applied Math., Russ. Acad. Sci., Moscow, 1996, no. 48.

  24. Sadov, S.Yu., Method of Averaging for the Equation of Satellite Oscillations at Eccentricity Close to Unity, in Materialy mezhdunar. konf. posvyashchennoi 175-letiyu P.L. Chebysheva (Proc. of Intern. Conf. Dedicated to 175 Years from P.L. Chebyshev’s Birth), Moscow: Mosk. Gos. Univ., 1996, vol. 2, pp. 305–307.

    Google Scholar 

  25. Sadov, S.Yu., Averaging of Degenerating Equation of Second Order, Int. Conf. on Some Topics in Mathematics (Samarkand, 1996). Abstracts, pp. 40–41.

  26. Sadov, S.Yu., Singular Normal Form for the Equation of Planar Oscillations of a Satellite, in Problemy nebesnoi mekhaniki. Tezisy dokladov (Issues of Celestial Mechanics. Abstracts), St. Petersburg: ITA RAN, 1997, pp. 58–60.

    Google Scholar 

  27. Sadov, S.Yu., Planar Motions of Almost Symmetrical Satellite Relative to Its Center of Mass with Rational Numbers of Rotation, Preprint of Keldysh Inst. of Applied Math., Russ. Acad. Sci., Moscow, 1997, no. 35.

  28. Sadov, S.Yu., Resonance Motions of a Satellite Relative to Its Center of Mass, Differential Equations, 1997, vol. 33, no. 6, p. 858.

    Google Scholar 

  29. Sadov, S.Yu., Singular Normal Form for a Quasilinear Ordinary Differential Equation, Nonlinear Analysis: Theory, Methods and Applications, 1997, vol. 30, no. 8, pp. 4973–4978.

    Article  MATH  MathSciNet  Google Scholar 

  30. Sadov, S.Yu., Functions That Determine Stability of Rational Rotations of a Near Symmetric Satellite, Mathematics and Computers in Simulations, 1998, vol. 45, nos. 5–6, pp. 465–484.

    MATH  MathSciNet  Google Scholar 

  31. Sadov, S.Yu., Chernous’ko Functions in the Problem of Stability of Rotation of a Satellite Relative to Its Center of Mass, 3 mezhd. simp. po klassich. i neb. mekh. Tezisy dokladov (3rd Intern. Symp. on Classical and Celestial Mechanics. Abstracts), Velikie Luki, 1998, pp. 56–57.

  32. Sadov, S.Yu., Lissajous Solutions of the Satellite Oscillation Equation: Stability and Bifurcations via Higher Order Averaging, Nonlinear Phenomena in Complex Systems, Dordrecht, Neth., 1999, vol. 2, no. 2, pp. 96–100.

    Google Scholar 

  33. Sarychev, V.A., Orientation of Artificial Satellites, Itogi Nauki Tekh., Ser.: Issled. Kosmich. Prostr., vol. 11, Moscow: VINITI, 1978.

    Google Scholar 

  34. Handbook of Mathematical Functions, Abramowitz, M. and Stegun, I.A., Eds., New York: Dover, 1971. Translated under the title Spravochnik po special’nym funktsiyam, Moscow: Nauka, 1979.

    Google Scholar 

  35. Wintner, A., Analiticheskie osnovy nebesnoi mekhaniki (Analytical Foundations of Celestial Mechanics), Moscow: Nauka, 1967.

    Google Scholar 

  36. Chernous’ko, F.L., Resonance Phenomena at Attitude Motion of Satellites, in Zh. Vych. Mat. i Mat. Fiz., 1963, vol. 3,issue 3, pp. 528–538.

    MathSciNet  Google Scholar 

  37. Fedoryuk, M.V., Metod perevala (Saddle-Point Method), Moscow: Nauka, 1977.

    Google Scholar 

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  1. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia

    S. Yu. Sadov

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Original Russian Text © S.Yu. Sadov, 2006, published in Kosmicheskie Issledovaniya, 2006, Vol. 44, No. 2, pp. 170–181.

An erratum to this article is available athttp://dx.doi.org/10.1134/S0010952506050108.

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Sadov, S.Y. Stability of resonance rotation of a satellite with respect to its center of mass in the orbit plane. Cosmic Res 44, 160–171 (2006). https://doi.org/10.1134/S0010952506020080

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  • DOI: https://doi.org/10.1134/S0010952506020080

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