Abstract
The first integrals of the Kepler problem are used to compute preliminary orbits starting from two short observed arcs of a celestial body, which may be obtained either by optical or by radar observations. We write polynomial equations for this problem, which can be solved using the powerful tools of computational Algebra. An algorithm to decide if the linkage of two short arcs is successful, i.e. if they belong to the same observed body, is proposed and tested numerically. This paper continues the research started in Gronchi et al. (Celest. Mech. Dyn. Astron. 107(3):299–318, 2010), where the angular momentum and the energy integrals were used. The use of a suitable component of the Laplace–Lenz vector in place of the energy turns out to be convenient, in fact the degree of the resulting system is reduced to less than half.
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References
Bini D.A.: Numerical computation of polynomial zeros by means of Aberth method. Numer. Algorithms 13(3–4), 179–200 (1997)
Celletti A., Negrini P.: Non-integrability of the problem of motion around an oblate planet. Celest. Mech. Dyn. Astron. 61, 253–260 (1995)
Farnocchia D., Tommei G., Milani A., Rossi A.: Innovative methods of correlation and orbit determination for space debris. Celest. Mech. Dyn. Astron. 107(1–2), 169–185 (2010)
Fujimoto K., Maruskin J.M., Scheeres D.J.: Circular and zero-inclination solutions for optical observations of Earth-orbiting objects. Celest. Mech. Dyn. Astron. 106, 157–182 (2010)
Gronchi G.F.: On the stationary points of the squared distance function between two ellipses with a common focus. SIAM Journ. Sci. Comp. 24(1), 61–80 (2002)
Gronchi G.F., Dimare L., Milani A.: Orbit determination with the two-body integrals. Celest. Mech. Dyn. Astron. 107(3), 299–318 (2010)
Milani A., Gronchi G.F.: Theory of Orbit Determination. Cambridge University Press, Cambridge, UK (2009)
Milani A., Gronchi G.F., Farnocchia D., Knežević Z., Jedicke R., Denneau L., Pierfederici F.: Topocentric orbit determination: algorithms for the next generation surveys. Icarus 195, 474–492 (2008)
Milani A., Sansaturio M.E., Chesley S.R.: The asteroid identification problem IV: attributions. Icarus 151, 150–159 (2001)
Milani, A., Tommei, G., Farnocchia, D., Rossi, A., Schildknecht, T., Jehn, R.: Orbit determination of space objects based on sparse optical data. ar**v:1012.5232 (2011)
Poincaré H.: Sur la détermination des orbites par la méthode de Laplace. Bull. Astron. 23, 161–187 (1906)
Roy A.E.: Orbital Motion. Institute of Physics Publishing, London (2005)
Taff L.G., Hall D.L.: The use of angles and angular rates. I—initial orbit determination. Celest. Mech. 16, 481–488 (1977)
Tommei G., Milani A., Rossi A.: Orbit determination of space debris: admissible regions. Celest. Mech. Dyn. Astron. 97(4), 289–304 (2007)
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Gronchi, G.F., Farnocchia, D. & Dimare, L. Orbit determination with the two-body integrals. II. Celest Mech Dyn Astr 110, 257–270 (2011). https://doi.org/10.1007/s10569-011-9357-z
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DOI: https://doi.org/10.1007/s10569-011-9357-z