Non-linear Post-Newtonian Equations for the Motion of Designated Targets with Respect to Space Based APT Laser Systems

  • Jose M. Gambi
  • Maria L. García del Pino
  • Maria C. Rodríguez-Teijeiro
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 26)


The equations introduced in this paper are aimed to gain accuracy in the determination of the motion of middle size space objects with respect to space based APT laser systems; therefore, they can be used for these systems to engage cataloged space debris objects whose size ranges between 1 and 10 cm. The equations are derived under the assumption that the framework of the Earth surrounding space is post-Newtonian and, unlike the standard p-N equations, they are valid for distant targets. Further, their time validity is also substantially larger than that of the standard equations. The reason is that they include non-linear terms that model the Earth tidal potential along the lines joining the systems and the targets. The equations are derived in local Cartesian orbital coordinates; therefore, they are primarily adapted for use with inertial-guided systems.


  1. 1.
    Ashby, N.: Relativity in the global positioning system. Living Rev. Relativ. 6, 1–42 (2003)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bahder, T.B.: Clock Synchronization and Navigation in the Vicinity of the Earth. Nova Science, New York (2009)Google Scholar
  3. 3.
    Curtis, H.D.: Orbital Mechanics for Engineering Students, 3rd edn., pp. 367–396. Elsevier, Oxford (2014)Google Scholar
  4. 4.
    Gambi, J.M., García del Pino, M.L.: Post-Newtonian Equations of Motion for Inertial Guided Space APT Systems. WSEAS Trans. Math. 14, 256–264 (2015). Available in
  5. 5.
    Gambi, J.M., Rodriguez-Teijeiro, M.C., García del Pino, M.L., Salas, M.: Shapiro time-delay within the Geolocation Problem by TDOA. IEEE Trans. Aerosp. Electron. Syst. 47(3), 1948–1962 (2011)Google Scholar
  6. 6.
    Gambi, J.M., Clares, J., García del Pino, M.L.: FDOA post-Newtonian equations for the location of passive emitters placed in the vicinity of the Earth. Aerosp. Sci. Technol. 46, 137–145 (2015)Google Scholar
  7. 7.
    Gambi, J.M., Rodriguez-Teijeiro, M.C., García del Pino, M.L.: Newtonian and post-Newtonian passive geolocation by TDOA. Aerosp. Sci. Technol. 51, 18–25 (2016)Google Scholar
  8. 8.
    Moritz, H., Hofmann-Wellenhof, B.: Geometry, Relativity, Geodesy. H. Wichmann Verlag GmbH, Berlin (1993)Google Scholar
  9. 9.
    Schmitz, M., Fasoulas, S., Utzmann, J.: Performance model for space-based laser debris sweepers. Acta Astronaut. 115, 376–386 (2015)CrossRefGoogle Scholar
  10. 10.
    Soffel, M.H.: Relativity in Astrometry, Celestial Mechanics and Geodesy. Springer, Berlin (1989)Google Scholar
  11. 11.
    Synge, J.L.: Relativity: The General Theory, pp. 87–95. North-Holland, Amsterdam (1960)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Jose M. Gambi
    • 1
  • Maria L. García del Pino
    • 2
  • Maria C. Rodríguez-Teijeiro
    • 3
  1. 1.Gregorio Millán InstituteUniv. Carlos III de MadridMadridSpain
  2. 2.Department of Mathematics, I.E.S. AlpajesAranjuez, MadridSpain
  3. 3.UNED-MadridMadridSpain

Personalised recommendations