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

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

EVALUATION OF ELECTRODE SURFACE TREATMENTS IN SLUDGE ELECTRO-OSMOSIS DEWATERING”

The drying of sludge produced by Wastewater Treatment Plants (WWTPs) is a very hard process due to the presence of the colloid fraction. Electro-osmosis could be a suitable technique to reduce the water content of the final sludge. Electrical fields of 10 V/cm, 15 V/cm and 20 V/cm have been studied for electro-osmosis tests in a static or dynamic apparatus, obtaining a dry solids content up to 40-45%, with respect to 25-30% obtained by mechanical methods. In order to optimise the apparatus, the corrosion behaviour of the anodic material appears the main critical aspect, due to the high circulating current density. Moreover, wear may be detrimental for the surface of rotating electrodes. We then investigated the behaviour of materials used as electrodes mainly by evaluating the efficiency of the process and their surface aspect after treatment. The full understanding of the electrochemical reactions developed at the anode are a key factor for the material choice. We compared the efficiency and the corrosion resistance of anodes made of titanium MMO with respect to bare stainless steel (AISI 304) and stainless steel coated by PVD technique with TiN, AlTiN and DLC. Characterization of the anodes samples by roughness tests and by AFM show that corrosion resistance of the DSA was the most suitable for our application. However, efficiencies of the electro-osmosis process for all the materials used have been found comparable, in terms of developed current densities and total energy consumptions, for low-test duration

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

ELECTRO-OSMOTIC DEWATERING OF SEWAGE SLUDGE: PRELIMINARY RESULTS

About half of the organic pollution load treated by the activated sludge process is oxidised and converted into water and carbon dioxide, while the remaining is converted into biomass, called “excess biological sludge” or “waste sludge”. At present, this technique is the cheapest way to remove colloidal and soluble organic pollutants from sewage, but it produces a huge amount of liquid waste sludge, with a dry solid (DS) content of less than 2%, rich in organic substances, mostly biodegradable. Therefore, it needs further processes to reduce its volume, by reducing its water content, and to reduce its polluting potential, due to its high content of biodegradable organic matter. However, sludge produced by Wastewater Treatment Plants (WWTPs) usually has a poor dewaterability. Industrially, mechanical dewatering is the method employed to increase the dry solids (DS) of sewage sludge. The aim of dewatering is the decrease of transport and disposal costs, but the efficiency of the mechanical dehydration methods is low. Electro-osmosis could be a suitable technique to reduce water content of the final sludge by the application of an electric field. Studies about electro-dewatering reported that the DS could increase to values around 40-45%, with respect to 25-30% obtained by mechanical methods, such as centrifuge, filter press and belt press. Aim Here, we investigated the parameters affecting pressure-driven electro-dewatering, with a static and a dynamic device, starting from types of sludge mechanically pre-dehydrated. We focused on the optimization of operating conditions to achieve the best results in terms of solids percentage at the minimum energy consumption. Methods Experiments have been performed by means of a lab-scale device able to produce both a mechanical pressure and an electric field. The use of both these systems could be efficient to increase the amount of dry solids in the final sludge. This device, as illustrated in Figure 1, consists of: • Cylindrical glass vessel (h=176 mm, Ø=80 mm) • Cooling water-jacket • Compressed air system (1-4.5 bar) • Double effect cylinder (200 mm stroke) SMC-CP96 (piston) • DC power supply (30 V-5 A) • Anode: DSA Ti (MMO), fixed to the piston • Cathode: stainless steel mesh (AISI 304) • Cloth: PTT (polytrimethyleneterephthalate) Figure 1-Equipment schematic of electro-dewatering device. Main Results The applied voltage has been set at 10 V, 15 V and 20 V and pressures at 3 bar. The results confirmed values of final DS around 40%. Initial DS and sludge conductivity, together with the applied electric field, are the key factors of electro-osmosis tests. The rotating piston highlighted that a mixing of sludge during the potential application could slightly increase the sludge conductivity, but cakes must be kept under 1.5 cm of thickness in order to achieve a good dewatering. In conclusion, in this paper we assess the feasibility of the electro-osmosis dewatering. It is a promising technique to remove water from sewage sludge, since it is able to increase DS of 15% on average with respect industrial mechanical dewatering. Moreover, this work proved that the treatment of a high DS cake is more difficult than a non-dewatered sludge due to a lower conductivity during the process

Innovative observing strategy and orbit determination for Low Earth Orbit Space Debris

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 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 has been proposed to 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 conclusion is 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, which would be observable by a reference radar system selected as comparison. It is also possible to maintain such a catalog within the accuracy requirements motivated by collision avoidance, and to detect catastrophic fragmentation events. However, such results depend upon specific assumptions on the sensor and on the software technologies

Orbit determination of space objects based on sparse optical data

While building up a catalog of Earth orbiting objects, if the available optical observations are sparse, not deliberate follow ups of specific objects, no orbit determination is possible without previous correlation of observations obtained at different times. This correlation step is the most computationally intensive, and becomes more and more difficult as the number of objects to be discovered increases. In this paper we tested two different algorithms (and the related prototype software) recently developed to solve the correlation problem for objects in geostationary orbit (GEO), including the accurate orbit determination by full least squares solutions with all six orbital elements. Because of the presence in the GEO region of a significant subpopulation of high area to mass objects, strongly affected by non-gravitational perturbations, it was actually necessary to solve also for dynamical parameters describing these effects, that is to fit between 6 and 8 free parameters for each orbit. The validation was based upon a set of real data, acquired from the ESA Space Debris Telescope (ESASDT) at the Teide observatory (Canary Islands). We proved that it is possible to assemble a set of sparse observations into a set of objects with orbits, starting from a sparse time distribution of observations, which would be compatible with a survey capable of covering the region of interest in the sky just once per night. This could result in a significant reduction of the requirements for a future telescope network, with respect to what would have been required with the previously known algorithm for correlation and orbit determination.Comment: 20 pages, 8 figure

Symplectic integration of space debris motion considering several Earth's shadowing models

In this work, we present a symplectic integration scheme to numerically compute space debris motion. Such an integrator is particularly suitable to obtain reliable trajectories of objects lying on high orbits, especially geostationary ones. Indeed, it has already been demonstrated that such objects could stay there for hundreds of years. Our model takes into account the Earth's gravitational potential, luni-solar and planetary gravitational perturbations and direct solar radiation pressure. Based on the analysis of the energy conservation and on a comparison with a high order non-symplectic integrator, we show that our algorithm allows us to use large time steps and keep accurate results. We also propose an innovative method to model Earth's shadow crossings by means of a smooth shadow function. In the particular framework of symplectic integration, such a function needs to be included analytically in the equations of motion in order to prevent numerical drifts of the energy. For the sake of completeness, both cylindrical shadows and penumbra transitions models are considered. We show that both models are not equivalent and that big discrepancies actually appear between associated orbits, especially for high area-to-mass ratios