Abstract
Synge’s equations for time-like geodesics in terms of Fermi coordinates are used to derive post-Newtonian equations for the relative motion of satellites in coplanar circular near orbits about the earth. The reference frame, co-moving with the base satellite, is assumed to be a Fermi frame, that is, inertial guided. The resulting system is autonomous, linear, and reduces to the equation of the geodesic deviation for nearby satellites. Hence, it can be used by some Acquisition, Pointing, and Tracking systems to increase the accuracy presently reached in locating passive radio-transmitters.
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Notes
- 1.
For any two events in E, \(P_{1}(x^{k_{1}})\), \(P_{2}(x^{k_{2}})\), for which there is a unique geodesic \(\varGamma _{P_{1}P_{2}}\) joining them with equations x k = ξ k(u) where u is an affine parameter ranging from 0 to 1, the world-function Ω(P 1, P 2) is defined by the line integral
$$\displaystyle{ \varOmega (P_{1},P_{2}) =\varOmega (x^{k_{1}},x^{k_{2}}) = \frac{1} {2}\int _{0}^{1}g_{ ij}U^{i}U^{j}du }$$taken along \(\varGamma _{P_{1}P_{2}}\) where \(x^{k_{1}} \equiv \xi ^{k}(0)\), \(x^{k_{2}} \equiv \xi ^{k}(1)\) and U k = dξ k∕du.
References
Guelma, M., Kogan, A., Kazarian, A., Livne, A., Orenstain, M., Michalik, H., Arnold, S.: Acquisition and pointing control for inter-satellite laser communications. IEEE Trans. Aerosp. Electron. Syst. 40(4), 1239–1248 (2004)
Norton, T., Conner, K., Covington, R., Ngo, H., Rink, C.: Development of reprogrammable high frame-rate detector devises for laser communication pointing, Acquisition and Tracking. IEEE Aerospace Conference Paper No. 1414 (2008). doi:10.1109/AERO.2008.4526342
Gambi, J.M., Rodriguez-Teijeiro, M.C., Garcia del Pino, M.L.: The post-Newtonian Geolocation problem by TDOA. In: Progress in Industrial Mathematics at ECMI 2010, p. 489. Springer, Berlin/Heidelberg (2012)
Synge, J.L.: Relativity: The General Theory, Chap. 2 North-Holland, New York (1960)
Adamy, D.L.: EW 102: A Second Course in Electronic Warfare. Artech House Radar Library, Chap. 6, p. 169. Horizon House Pub. Inc., Boston (2004)
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Gambi, J.M., del Pino, M.L.G., Tung, M.M. (2016). Post-Newtonian Orbital Equations for Fermi Frames in the Vicinity the Earth. In: Russo, G., Capasso, V., Nicosia, G., Romano, V. (eds) Progress in Industrial Mathematics at ECMI 2014. ECMI 2014. Mathematics in Industry(), vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-23413-7_129
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DOI: https://doi.org/10.1007/978-3-319-23413-7_129
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