Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
We make the speed of light, c, and the universal gravitational constant, G, equal to 1, so that all magnitudes with dimension are given in seconds. Thus, m, the mass of the Earth, which is 1.479 × 10−11 approx., and the semi-major axes of the debris objects e.g. in LEO orbits, which range between 2.258 × 10−2 and 2.792 × 10−2 approx., give that ε is of the order of 10−9.
References
Ashby, N.: Relativity in the global positioning system. Living Rev. Relativ. 6, 1–42 (2003)
Bahder, T.B.: Clock Synchronization and Navigation in the Vicinity of the Earth. Nova Science, New York (2009)
Curtis, H.D.: Orbital Mechanics for Engineering Students, 3rd edn., pp. 367–396. Elsevier, Oxford (2014)
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 http://www.wseas.org/multimedia/journals/mathematics/2015/a4872609-118.pdf
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)
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)
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)
Moritz, H., Hofmann-Wellenhof, B.: Geometry, Relativity, Geodesy. H. Wichmann Verlag GmbH, Berlin (1993)
Schmitz, M., Fasoulas, S., Utzmann, J.: Performance model for space-based laser debris sweepers. Acta Astronaut. 115, 376–386 (2015)
Soffel, M.H.: Relativity in Astrometry, Celestial Mechanics and Geodesy. Springer, Berlin (1989)
Synge, J.L.: Relativity: The General Theory, pp. 87–95. North-Holland, Amsterdam (1960)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Gambi, J.M., García del Pino, M.L., Rodríguez-Teijeiro, M.C. (2017). Non-linear Post-Newtonian Equations for the Motion of Designated Targets with Respect to Space Based APT Laser Systems. In: Quintela, P., et al. Progress in Industrial Mathematics at ECMI 2016. ECMI 2016. Mathematics in Industry(), vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-63082-3_73
Download citation
DOI: https://doi.org/10.1007/978-3-319-63082-3_73
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-63081-6
Online ISBN: 978-3-319-63082-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)