Orbit determination for standard-like maps: asymptotic expansion of the confidence region in regular zones
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(2020) 132:40
ORIGINAL ARTICLE
Orbit determination for standard-like maps: asymptotic expansion of the confidence region in regular zones Stefano Marò1 Received: 17 February 2020 / Revised: 11 June 2020 / Accepted: 6 August 2020 © Springer Nature B.V. 2020
Abstract We deal with the orbit determination problem for a class of maps of the cylinder generalizing the Chirikov standard map. The problem consists of determining the initial conditions and other parameters of an orbit from some observations. A solution to this problem goes back to Gauss and leads to the least squares method. Since the observations admit errors, the solution comes with a confidence region describing the uncertainty of the solution itself. We study the behavior of the confidence region in the case of a simultaneous increase in the number of observations and the time span over which they are performed. More precisely, we describe the geometry of the confidence region for solutions in regular zones. We prove an estimate of the trend of the uncertainties in a set of positive measure of the phase space, made of invariant curve. Our result gives an analytical proof of some known numerical evidences. Keywords Orbit determination · Confidence region · Invariant curves
1 Introduction Orbit determination is a classical problem in applied Celestial Mechanics. It consists of recovering information on some parameters (initial conditions or dynamical parameters) of a model, given some observations. The first notable result was obtained by Gauss in the nineteenth century (Gauss 1809/1963). He was able to recover the orbit of Ceres given the 21 observations made by Piazzi in different nights. Gauss method was composed of two steps. First, an approximation of the solution was computed, and then, the least squares method was applied to improve the first approximation. This strategy is still in use nowadays, and the applications have become wide-ranging. The accurate determination of orbits of NEOs is essential in the impact monitoring activity. On the other hand, the targets of many space missions include the determination of some
This work was supported by the National Group of Mathematical Physics (GNFM-INdAM) through the project “Orbit Determination: from order to chaos” (Progetto Giovani 2019).
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Stefano Marò [email protected] Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo, 5, Pisa, Italy 0123456789().: V,-vol
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unknown parameter. Typical examples are the ESA/JAXA BepiColombo mission to Mercury, the NASA JUNO and ESA JUICE missions to Jupiter (see Lari and Milani 2019). The result of an orbit determination process (called nominal solution) always comes with a confidence region, summarizing the uncertainties of the result itself. Its behavior is of crucial importance in applied problems; for examples, it is at the base of the definition of the impact probability in impact monitoring (Milani and Valsecchi 1999). Hence, it is important to study the confidence region as the number of observations
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