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Analytical and Semianalytical Propagation of Space Orbits: The Role of Polar-Nodal Variables

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Part of the book series: Astrophysics and Space Science Proceedings ((ASSSP,volume 44))

Abstract

The role of different sets of variables in analytical or semianalytical solutions to the artificial satellite theory is reviewed. In particular, it is shown that using polar-nodal variables or non-singular variables based on them is clearly advantageous in the evaluation of the transformation from mean to osculating elements.

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Notes

  1. 1.

    https://www.orekit.org/, http://tastrody.unirioja.es/dsst (accessed October 22, 2015).

References

  • Aksenov, Y.P., Grebenikov, Y.A., Demin, V.G.: General solution of the problem of the motion of an artificial satellite in the normal field of the earth’s attraction. Planet. Space Sci. 9 (8), 491–498 (1962)

    Article  ADS  Google Scholar 

  • Aksnes, K.: A note on ‘The main problem of satellite theory for small eccentricities, by A. Deprit and A. Rom, 1970’. Celest. Mech. 4, 119–121 (1971)

    Google Scholar 

  • Aksnes, K.: On the use of the Hill variables in artificial satellite theory. Astron. Astrophys. 17, 70–75 (1972)

    ADS  MATH  Google Scholar 

  • Alfriend, K.T., Coffey, S.L.: Elimination of the perigee in the satellite problem. Celest. Mech. 32 (2), 163–172 (1984)

    Article  ADS  MATH  Google Scholar 

  • Armellin, R., San-Juan, J.F., Lara, M.: End-of-life disposal of high elliptical orbit missions: the case of INTEGRAL. Adv. Space Res. 56 (3), 479–493 (2015)

    Article  ADS  Google Scholar 

  • Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Springer, New York (1989)

    Book  Google Scholar 

  • Brouwer, D.: Solution of the problem of artificial satellite theory without drag. Astron. J. 64, 378–397 (1959)

    Article  ADS  MathSciNet  Google Scholar 

  • Brouwer, D., Clemence, G.M.: Methods of Celestial Mechanics. Academic, New York (1961)

    MATH  Google Scholar 

  • Cid, R., Lahulla, J.F.: Perturbaciones de corto periodo en el movimiento de un satélite artificial en función de las variables de Hill. Publ. Rev. Acad. Cienc. Zaragoza 24, 159–165 (1969)

    Google Scholar 

  • Cid, R., Lahulla, J.F.: Perturbaciones de segundo orden y corto periodo, para el movimiento de un satélite artificial, en las variables de Hill. Publ. Rev. Acad. Cienc. Zaragoza 26 (2), 333–343 (1971)

    Google Scholar 

  • Cid, R., Ferrer, S., Sein-Echaluce, M.L.: On the radial intermediaries and the time transformation in satellite theory. Celest. Mech. 38 (2), 191–205 (1986)

    Article  ADS  MATH  Google Scholar 

  • Coffey, S., Deprit, A.: Third-order solution to the main problem in satellite theory. J. Guid. Control Dyn. 5 (4), 366–371 (1982)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Coffey, S.L., Deprit, A., Miller, B.R.: The critical inclination in artificial satellite theory. Celest. Mech. 39 (4), 365–406 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Danielson, D.A., Sagovac, C.P., Neta, B., Early, L.W.: Semianalytic Satellite Theory. Technical Report NPS-MA-95-002, Naval Postgraduate School, Monterey, CA. Dep of Math (1995)

    Google Scholar 

  • Delaunay, C.E.: La Théorie du Mouvement de la Lune, Premier volume. Mem. Acad. Sci. Inst. Imp. Fr., vol. 28. Mallet-Bachellier, Paris (1860)

    Google Scholar 

  • Deprit, A.: Canonical transformations depending on a small parameter. Celest. Mech. 1 (1), 12–30 (1969)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Deprit, A.: The elimination of the parallax in satellite theory. Celest. Mech. 24 (2), 111–153 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Deprit, A., Ferrer, S.: Simplifications in the theory of artificial satellites. J. Astronaut. Sci. 37, 451–463 (1989)

    ADS  MathSciNet  Google Scholar 

  • Deprit, A., Rom, A.: The main problem of artificial satellite theory for small and moderate eccentricities. Celest. Mech. 2 (2), 166–206 (1970)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Deprit, E., Deprit, A.: Poincaré’s méthode nouvelle by skew composition. Celest. Mech. Dyn. Astron. 74, 175–197 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Ferrer, S., Lara, M.: Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. J. Geom. Mech. 2 (3), 223–241 (2010)

    MathSciNet  MATH  Google Scholar 

  • Garfinkel, B.: The orbit of a satellite of an oblate planet. Astron. J. 64 (9), 353–367 (1959)

    Article  ADS  MathSciNet  Google Scholar 

  • Garfinkel, B.: Formal solution in the problem of small divisors. Astron. J. 71, 657–669 (1966)

    Article  ADS  MathSciNet  Google Scholar 

  • Golikov, A.R.: THEONA—a numerical-analytical theory of motion of artificial satellites of celestial bodies. Cosm. Res. 50 (6), 449–458 (2012)

    Article  ADS  Google Scholar 

  • Gurfil, P., Lara, M.: Satellite onboard orbit propagation using Deprit’s radial intermediary. Celest. Mech. Dyn. Astron. 120, 217–232 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Henrard, J.: Virtual singularities in the artificial satellite theory. Celest. Mech. 10 (4), 437–449 (1974)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Henrard, J., Wauthier, P.: A geometric approach to the ideal resonance problem. Celest. Mech. 44, 227–238 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Hoots, F.R.: Reformulation of the Brouwer geopotential theory for improved computational efficiency. Celest. Mech. 24, 367–375 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Hoots, F.R., Roehrich, R.L.: Models for Propagation of the NORAD Element Sets. Project SPACETRACK, Rep 3, US Air Force Aerosp Def Command, Colorado Springs, CO (1980)

    Google Scholar 

  • Hori, G.: Theory of general perturbation with unspecified canonical variables. Publ. Astron. Soc. Jpn. 18 (4), 287–296 (1966)

    ADS  Google Scholar 

  • Izsak, I.G.: A note on perturbation theory. Astron. J. 68, 559–561 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  • Jupp, A.H.: A second-order solution of the ideal resonance problem by Lie series. Celest. Mech. 5, 8–26 (1972)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Jupp, A.H.: The problem of the critical inclination revisited. Celest. Mech. 11 (3), 361–378 (1975)

    Article  ADS  MATH  Google Scholar 

  • Kaula, W.M.: Theory of Satellite Geodesy. Applications of Satellites to Geodesy. Blaisdell, Waltham (1966)

    MATH  Google Scholar 

  • Kelso, T.J.: Analysis of the Iridium 33–Cosmos 2251 collision. In: Paper AAS 09-368, Am. Astronaut. Soc. (2009)

    Google Scholar 

  • Kinoshita, H.: Third-Order Solution of an Artificial-Satellite Theory. SAO Special Rep 379 (1977)

    Google Scholar 

  • Kislik, M.D.: The path of an artificial earth satellite in the normal gravitational field of the earth. Planet. Space Sci. 8, 86–96 (1961)

    Article  ADS  Google Scholar 

  • Kozai, Y.: The motion of a close earth satellite. Astron. J. 64 (11), 367–377 (1959)

    Article  ADS  MathSciNet  Google Scholar 

  • Kozai, Y.: Second-order solution of artificial satellite theory without air drag. Astron. J. 67 (7), 446–461 (1962)

    Article  ADS  MathSciNet  Google Scholar 

  • Lara, M.: Efficient formulation of the periodic corrections in Brouwer’s gravity solution. Math. Probl. Eng. 2015, Article ID 980652 (2015a)

    Article  MathSciNet  Google Scholar 

  • Lara, M.: On inclination resonances in artificial satellite theory. Acta Astronaut. 110, 239–246 (2015b)

    Article  ADS  Google Scholar 

  • Lara, M., San-Juan, J.F., López-Ochoa, L.M.: Averaging tesseral effects: closed form relegation versus expansions of elliptic motion. Math. Probl. Eng. 2013, Article ID 570127 (2013)

    MathSciNet  Google Scholar 

  • Lara, M., San-Juan, J.F., López-Ochoa, L.M.: Delaunay variables approach to the elimination of the perigee in artificial satellite theory. Celest. Mech. Dyn. Astron. 120, 39–56 (2014a)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Lara, M., San-Juan, J.F., López-Ochoa, L.M.: Proper averaging via parallax elimination. Adv. Astronaut. Sci. 150, 315–331 (2014b)

    Google Scholar 

  • Lara, M., de Moraes, R.V., Sanchez, D.M., Prado, A.F.B.A.: Efficient computation of short-period analytical corrections due to third-body effects. Adv. Astronaut. Sci. 155, 437–455 (2015)

    Google Scholar 

  • Lyddane, R.H.: Small eccentricities or inclinations in the Brouwer theory of the artificial satellite. Astron. J. 68, 555–558 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  • McClain, W.D.: A Recursively Formulated First-Order Semianalytic Artificial Satellite Theory Based on the Generalized Method of Averaging, Volume 1: The Generalized Method of Averaging Applied to the Artificial Satellite Problem, 2nd edn. NASA CR-156782. NASA, Greenbelt (1977)

    Google Scholar 

  • Mitropolsky, I.: Averaging method in non-linear mechanics. Int. J. Non Linear Mech. 2, 69–96 (1967)

    Article  ADS  MathSciNet  Google Scholar 

  • Osácar, C., Palacián, J.F.: Decomposition of functions for elliptical orbits. Celest. Mech. Dyn. Astron. 60, 207–223 (1994)

    Article  ADS  MATH  Google Scholar 

  • Poincaré, H.: Les methodes nouvelles de la mecanique celeste. Gauthier-Villars, Paris (1892)

    MATH  Google Scholar 

  • Sanders, J.A., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems, 2nd edn. Springer, New York (2007)

    MATH  Google Scholar 

  • Setty, S.J., Cefola, P.J., Montenbruck, O., Fiedler, H., Lara, M.: Investigating the suitability of analytical and semi-analytical satellite theories for space object catalogue maintenance in geosynchronous regime. Adv. Astronaut. Sci. 150, 1027–1042 (2014)

    Google Scholar 

  • Vallado, D.A., Crawford, P., Hujsak, R., Kelso, T.S.: Revisiting spacetrack report #3. In: Paper AIAA 2006-6753, Am. Inst. Aeronaut. Astronaut. (2006)

    Google Scholar 

  • Vinti, J.P.: New method of solution for unretarded satellite orbits. J. Res. Natl. Bur. Stand. 62B (2), 105–162 (1959)

    Article  MathSciNet  Google Scholar 

  • von Zeipel, H.: Research on the motion of minor planets (recherches sur le mouvement des petites planètes). NASA Transl: NASA TT F-9445 (1965) (1916, 1917, 1918).

    Google Scholar 

  • Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 1st edn. Cambridge University Press, Cambridge (1904)

    MATH  Google Scholar 

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Acknowledgements

Partial support from projects ESP2013-41634-P and ESP2014-57071-R of the Ministry of Economic Affairs and Competitiveness of Spain is recognized.

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Correspondence to Martin Lara .

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Lara, M. (2016). Analytical and Semianalytical Propagation of Space Orbits: The Role of Polar-Nodal Variables. In: Gómez, G., Masdemont, J. (eds) Astrodynamics Network AstroNet-II. Astrophysics and Space Science Proceedings, vol 44. Springer, Cham. https://doi.org/10.1007/978-3-319-23986-6_11

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