The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities

We study the long term evolution of the distance between two Keplerian confocal trajectories in the framework of the averaged restricted 3-body problem. The bodies may represent the Sun, a solar system planet and an asteroid. The secular evolution of the orbital elements of the asteroid is computed by averaging the equations of motion over the mean anomalies of the asteroid and the planet. When an orbit crossing with the planet occurs the averaged equations become singular. However, it is possible to define piecewise differentiable solutions by extending the averaged vector field beyond the singularity from both sides of the orbit crossing set. In this paper we improve the previous results, concerning in particular the singularity extraction technique, and show that the extended vector fields are Lipschitz-continuous. Moreover, we consider the distance between the Keplerian trajectories of the small body and of the planet. Apart from exceptional cases, we can select a sign for this distance so that it becomes an analytic map of the orbital elements near to crossing configurations. We prove that the evolution of the 'signed' distance along the averaged vector field is more regular than that of the elements in a neighborhood of crossing times. A comparison between averaged and non-averaged evolutions and an application of these results are shown using orbits of near-Earth asteroids.Comment: 29 pages, 8 figure

Long term dynamics for the restricted N-body problem with mean motion resonances and crossing singularities

We consider the long term dynamics of the restricted N -body problem, modeling in a statistical sense the motion of an asteroid in the gravitational field of the Sun and the solar system planets. We deal with the case of a mean motion resonance with one planet and assume that the osculating trajectory of the asteroid crosses the one of some planet, possibly different from the resonant one, during the evolution. Such crossings produce singularities in the differential equations for the motion of the asteroid, obtained by standard perturbation theory. In this work we prove that the vector field of these equations can be extended to two locally Lipschitz-continuous vector fields on both sides of a set of crossing conditions. This allows us to define generalized solutions, continuous but not differentiable, going beyond these singularities. Moreover, we prove that the long term evolution of the ’signed’ orbit distance (Gronchi and Tommei 2007) between the asteroid and the planet is differentiable in a neighborhood of the crossing times. In case of crossings with the resonant planet we recover the known dynamical protection mechanism against collisions. We conclude with a numerical comparison between the long term and the full evolutions in the case of asteroids belonging to the ’Alinda’ and ’Toro’ classes (Milani et al. 1989). This work extends the results in (Gronchi and Tardioli 2013) to the relevant case of asteroids in mean motion resonance with a planet

Maps of secular resonances in the NEO region

Context. From numerical simulations, it is known that some secular resonances may affect the motion of near-Earth objects (NEOs). However, the specific location of the secular resonance inside the NEO region is not fully known, because the methods previously used to predict their location can not be used for highly eccentric orbits and when the NEOs cross the orbits of the planets. Aims. In this paper, we aim to map the secular resonances with the planets from Venus to Saturn in the NEO region, even for high values of the eccentricity. Methods. We used an averaged semi-analytical model that can deal with orbit crossing singularities for the computation of the secular dynamics of NEOs, from which we can obtain suitable proper elements and proper frequencies. Then, we computed the proper frequencies over a uniform grid in the proper elements space. Secular resonances are thus located by the level curves corresponding to the proper frequencies of the planets. Results. We determined the location of the secular resonances with the planets from Venus to Saturn, showing that they appear well inside the NEO region. By using full numerical N-body simulations we also showed that the location predicted by our method is fairly accurate. Finally, we provided some indications about possible dynamical paths inside the NEO region, due to the presence of secular resonances.Comment: Accepted for publication in A&

Orbit Determination with the two-body Integrals

We investigate a method to compute a finite set of preliminary orbits for solar system bodies using the first integrals of the Kepler problem. This method is thought for the applications to the modern sets of astrometric observations, where often the information contained in the observations allows only to compute, by interpolation, two angular positions of the observed body and their time derivatives at a given epoch; we call this set of data attributable. Given two attributables of the same body at two different epochs we can use the energy and angular momentum integrals of the two-body problem to write a system of polynomial equations for the topocentric distance and the radial velocity at the two epochs. We define two different algorithms for the computation of the solutions, based on different ways to perform elimination of variables and obtain a univariate polynomial. Moreover we use the redundancy of the data to test the hypothesis that two attributables belong to the same body (linkage problem). It is also possible to compute a covariance matrix, describing the uncertainty of the preliminary orbits which results from the observation error statistics. The performance of this method has been investigated by using a large set of simulated observations of the Pan-STARRS project.Comment: 23 pages, 1 figur

Sleep and psychological characteristics in habitual self-awakeners and forced awakeners.

Previous studies described the modifications of physiological and behavioural variables associated with self-awakening, while only few studies assessed subjective sleep quality and psychological characteristics in habitual self-awakeners. Here we investigated self-reported sleep habits and features, as well as psychological variables of habitual self-awakeners and forced-awakeners, with special regard to subjective sleep quality, personality characteristics, anxiety and depression symptoms. In our sample, the prevalence of habitual self-awakeners was 15.1%. Compared to forced-awakeners, habitual self-awakeners showed more regular sleep/wake schedules and were more frequently morning types. Moreover, habitual self-awakeners referred to be more satisfied about their sleep, to wake up more easily in the morning, to need less time to get out of bed and to feel more refreshed upon awakening than forced-awakeners. We also observed an association between the habit of self-awakening and the "ability" to set the awakening to an unusual time. Concerning psychological features, habitual self-awakeners showed higher scores in Conscientiousness and Openness and lower scores in Extraversion compared to forced-awakeners, whereas no differences between groups emerged for anxiety and depression levels. In conclusion, our findings point to an association between the habit of self-awakening and good subjective sleep quality. In this perspective, future research should objectively test in detail the effects of the self-awakening habit on sleep structure and organization, taking into account also microstructural sleep features

False memories formation is increased in individuals with insomnia

Previous studies suggest that sleep can influence false memories formation. Specifically, acute sleep loss has been shown to promote false memories production by impairing memory retrieval at subsequent testing. Surprisingly, the relationship between sleep and false memories has only been investigated in healthy subjects but not in individuals with insomnia, whose sleep is objectively impaired compared to healthy subjects. Indeed, this population shows several cognitive impairments involving prefrontal functioning that could affect source monitoring processes and contribute to false memories generation. Moreover, it has been previously reported that subjects with insomnia differentially process sleep-related versus neutral stimuli. Therefore, the aim of the present study was to compare false memories production between individuals with insomnia symptoms and good sleepers, and to evaluate the possible influence of stimulus category (neutral versus sleep-related) in the two groups. The results show that false memories are globally increased in participants reporting insomnia symptoms compared to good sleepers. A reduction in source monitoring ability was also observed in the former group, suggesting that an impairment of this executive function could be especially involved in false memories formation. Moreover, our data seem to confirm that false memories production in individuals with insomnia symptoms appears significantly modulated by stimulus category

Orbit Determination with the two-body Integrals. II

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 radar observations. We write polynomial equations for this problem, that we can solve 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. In this paper we continue the research started in [Gronchi, Dimare, Milani, 'Orbit determination with the two-body intergrals', CMDA (2010) 107/3, 299-318], where the angular momentum and the energy integrals were used. 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.Comment: 15 pages, 4 figure

Innovative system of very wide field optical sensors for space surveillance in the LEO region

ABSTRACT We present the results of a large scale simulation, reproducing the behavior of a data center for the build-up and maintenance of a complete catalog of space debris in the upper part of the low Earth orbits region (LEO). The purpose is to determine the achievable performances of a network of advanced optical sensors, through the use of the newest orbit determination algorithms developed by the Department of Mathematics of Pisa (DM). Such a network was designed and proposed to the European Space Agency (ESA) in the Space Situational Awareness (SSA) framework by Carlo Gavazzi Space SpA (CGS), Istituto Nazionale di Astrofisica (INAF), DM and Istituto di Scienza e Tecnologie dell'Informazione (ISTI-CNR). The latest developed orbit determination algorithms were used to process simulated observations from the proposed network. In particular two innovative methods for preliminary orbit determination based on the first integrals of the Kepler problem were compared, by using them to process the same data. In both cases, the results showed that it is possible to use a network of optical sensors to build up a catalog containing more than 98% of the objects with perigee height between 1100 and 2000 km, and diameter greater than 8 cm. Such a catalog is obtained in just two months of observations. However, such results depend upon specific assumptions on the sensor and on the software technologies