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Shadowing Lemma and chaotic orbit determination
No full textOrbit determination is possible for a chaotic orbit of a dynamical system, given a
finite set of observations, provided the initial conditions are at the central time. The Shadowing
Lemma (Anosov 1967; Bowen in J Differ Equ 18:333–356, 1975) can be seen as a way to
connect the orbit obtained using the observations with a real trajectory.Anorbit is a shadowing
of the trajectory if it stays close to the real trajectory for some amount of time. In a simple
discrete model, the standard map, we tackle the problem of chaotic orbit determination when
observations extend beyond the predictability horizon. If the orbit is hyperbolic, a shadowing
orbit is computed by the least squares orbit determination. We test both the convergence
of the orbit determination iterative procedure and the behaviour of the uncertainties as a
function of the maximum number of map iterations observed. When the initial conditions
belong to a chaotic orbit, the orbit determination is made impossible by numerical instability
beyond a computability horizon, which can be approximately predicted by a simple formula.
Moreover, the uncertainty of the results is sharply increased if a dynamical parameter is
added to the initial conditions as parameter to be estimated. The Shadowing Lemma does
not dictate what the asymptotic behaviour of the uncertainties should be. These phenomena
have significant implications, which remain to be studied, in practical problems of orbit
determination involving chaos, such as the chaotic rotation state of a celestial body and a
chaotic orbit of a planet-crossing asteroid undergoing many close approaches
Asteroid family ages
No full textA new family classification, based on a catalog of proper elements with similar to 384,000 numbered asteroids and on new methods is available. For the 45 dynamical families with >250 members identified in this classification, we present an attempt to obtain statistically significant ages: we succeeded in computing ages for 37 collisional families.
We used a rigorous method, including a least squares fit of the two sides of a V-shape plot in the proper semimajor axis, inverse diameter plane to determine the corresponding slopes, an advanced error model for the uncertainties of asteroid diameters, an iterative outlier rejection scheme and quality control. The best available Yarkovsky measurement was used to estimate a calibration of the Yarkovsky effect for each family. The results are presented separately for the families originated in fragmentation or cratering events, for the young, compact families and for the truncated, one-sided families. For all the computed ages the corresponding uncertainties are provided, and the results are discussed and compared with the literature. The ages of several families have been estimated for the first time, in other cases the accuracy has been improved. We have been quite successful in computing ages for old families, we have significant results for both young and ancient, while we have little, if any, evidence for primordial families. We found 2 cases where two separate dynamical families form together a single V-shape with compatible slopes, thus indicating a single collisional event. We have also found 3 examples of dynamical families containing multiple collisional families, plus a dubious case: for these we have obtained discordant slopes for the two sides of the V-shape, resulting in distinct ages. We have found 2 cases of families containing a conspicuous subfamily, such that it is possible to measure the slope of a distinct V-shape, thus the age of the secondary collision. We also provide data on the central gaps appearing in some families. The ages computed in this paper are obtained with a single and uniform methodology, thus the ages of different families can be compared, providing a first example of collisional chronology of the asteroid main belt. (C) 2015 Elsevier Inc. All rights reserved
On the ages of resonant, eroded and fossil asteroid families
Get PDFIn this work we have estimated 10 collisional ages of 9 families for which for different reasons our previous attempts failed. In general, these are difficult cases that required dedicated effort, such as a new family classifications for asteroids in mean motion resonances, in particular the 1/1 and 2/1 with Jupiter, as well as a revision of the classification inside the 3/2 resonance. Of the families locked in mean motion resonances, by employing a numerical calibration to estimate the Yarkovsky effect in proper eccentricity, we succeeded in determining ages of the families of (1911) Schubart and of the super-Hilda family, assuming this is actually a severely eroded original family of (153) Hilda. In the Trojan region we found families with almost no Yarkovsky evolution, for which we could compute only physically implausible ages. Hence, we interpreted their modest dispersions of proper elements as implying that the Trojan asteroid families are fossil families, frozen at their proper elements determined by the original ejection velocity field. We have found a new family, among the Griquas locked in the 2/1 resonance with Jupiter, the family of (11097) 1994 UD1. We have estimated the ages of 6 families affected by secular resonances: families of (5) Astraea, (25) Phocaea, (283) Emma, (363) Padua, (686) Gersuind, and (945) Barcelona. By using in all these cases a numerical calibration method, we have shown that the secular resonances do not affect significantly the secular change of proper a. We have confirmed the existence of the family resulting from cratering on (5) Astraea by computing a new set of resonant proper elements adapted to the resonance g+g5-2g6: this new family has a much larger membership and has a shape compatible with simple collisional models. For the family of (145) Adeona we could estimate the age only after removal of a number of assumed interlopers. With the present paper we have concluded the series dedicated to the determination of asteroid ages with a uniform method. Overall we computed 53 ages for a total of 49 families. For the future work there remain families too small at present to provide reliable estimates, as well as some complex families (221, 135, 298) which may have more ages than we could currently estimate. Future improvement of some already determined family ages is also possible by increasing family membership, revising the calibrations, and using more reliable physical data
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