ABSTRACT The purpose of this paper is to find a set of canonical elements to use within the frame... more ABSTRACT The purpose of this paper is to find a set of canonical elements to use within the framework of Öpik theory of close encounters of a small body with a planet (Öpik, Interplanetary Encounters, 1976). Since the small body travels along a planetocentric hyperbola during the close approach and Öpik formulas are valid, without approximations, only at collision, we derive a set of canonical elements for hyperbolic collision orbits (eccentricity e → 1+, semi-major axis a fixed) and then we introduce the unperturbed velocity of the small body and the distance covered along the asymptote as a new canonically conjugate pair of orbital elements. An interesting result would be to get a canonical set containing the coordinates in the Target Plane (TP), useful for the analysis of the future encounters: in the last part we prove that this is not possible.
The main problem in the orbit determination of the space debris population is the correlation of ... more The main problem in the orbit determination of the space debris population is the correlation of independently ob- served tracklets, that are sets of observations over a short time. The information contained in such data are not suf- ficient for a complete determination of an orbit, thus we need to find two or more tracklets belonging to the same physical object and an orbit fitting all the observations. In this paper we will show how to use the admissible re- gion tool to generate a set of virtual objects (Virtual De- bris) which can be used as alternate preliminary orbits and as starting points for a recursive procedure of corre- lation. We shall focus on optical observations of GEO showing some preliminary results testing the 2007 data by the ESA Optical Ground Station (OGS) at Teide Ob- servatory (Canary Islands), after an astrometric reductio n performed by University of Bern.
ABSTRACT BepiColombo (BC) is an ESA mission, including two satellites orbiting Mercury, to be lau... more ABSTRACT BepiColombo (BC) is an ESA mission, including two satellites orbiting Mercury, to be launched in 2015. JUNO is a NASA mission, already launched and on the way to Jupiter. Both S/C carry dedicated Ka - band transponders and other equipment to perform the most advanced interplanetary radio science experiments made possible by state of the art technology. A consistent set of models compatible with the accuracy of the range and range - rate measurements needs to include the dynamic model, the observation model, and the appropriate spacetime reference and coordinate systems. For BC the model for mercury - centric orbit has a limited need for relativistic terms, but the independent variable of the equations of motion needs to be a form of Dynamical Mercury Time, wit h comparatively large nonlinear differences with respect to the corresponding time coordinate with respect to the solar system barycenter (SSB). The dynamical model for the orbits of the planets Mercury and Earth need a complete PPN formulation and a large set of asteroids considered as perturbing masses; the very precise definition of the SSB is a challenge. The observation model requires as many as four separate iterations, plus a Shapiro term with second order corrections. For JUNO the dynamic model o f the jovicentric orbit requires direct and tidal perturbations from many satellites, and Jupiter relativistic terms including Lens - Thirring effect; the independent variable must be a Dynamical Jupiter Time. The observations model needs to include a Jupiter Shapiro effect, including an oblateness term.
Monthly Notices of the Royal Astronomical Society, 2014
Juno is a NASA mission launched in 2011 with the goal of studying Jupiter. The probe will arrive ... more Juno is a NASA mission launched in 2011 with the goal of studying Jupiter. The probe will arrive to the planet in 2016 and will be placed for one year in a polar higheccentric orbit to study the composition of the planet, the gravity and the magnetic field. The Italian Space Agency (ASI) provided the radio science instrument KaT (Ka-Band Translator) used for the gravity experiment, which has the goal of studying the Jupiter's deep structure by mapping the planet's gravity: such instrument takes advantage of synergies with a similar tool in development for BepiColombo, the ESA cornerstone mission to Mercury. The Celestial Mechanics Group of the University of Pisa, being part of the Juno Italian team, is developing an orbit determination and parameters estimation software for processing the real data independently from NASA software ODP. This paper has a twofold goal: first, to tell about the development of this software highlighting the models used, second, to perform a sensitivity analysis on the parameters of interest to the mission.
Proceedings of the International Astronomical Union, 2006
The distance between two confocal Keplerian orbits, called MOID in the literature, is a useful to... more The distance between two confocal Keplerian orbits, called MOID in the literature, is a useful tool to know if two celestial bodies can collide or undergo a very close approach. Two confocal orbits may get close at more than one pair of points, thus it is useful to compute not only the absolute minimum of the distance function d between
Monthly Notices of the Royal Astronomical Society, 2011
While building up a catalog of Earth orbiting objects, if the available optical observations are ... more 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 nongravitational 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.
ABSTRACT The purpose of this paper is to find a set of canonical elements to use within the frame... more ABSTRACT The purpose of this paper is to find a set of canonical elements to use within the framework of Öpik theory of close encounters of a small body with a planet (Öpik, Interplanetary Encounters, 1976). Since the small body travels along a planetocentric hyperbola during the close approach and Öpik formulas are valid, without approximations, only at collision, we derive a set of canonical elements for hyperbolic collision orbits (eccentricity e → 1+, semi-major axis a fixed) and then we introduce the unperturbed velocity of the small body and the distance covered along the asymptote as a new canonically conjugate pair of orbital elements. An interesting result would be to get a canonical set containing the coordinates in the Target Plane (TP), useful for the analysis of the future encounters: in the last part we prove that this is not possible.
The main problem in the orbit determination of the space debris population is the correlation of ... more The main problem in the orbit determination of the space debris population is the correlation of independently ob- served tracklets, that are sets of observations over a short time. The information contained in such data are not suf- ficient for a complete determination of an orbit, thus we need to find two or more tracklets belonging to the same physical object and an orbit fitting all the observations. In this paper we will show how to use the admissible re- gion tool to generate a set of virtual objects (Virtual De- bris) which can be used as alternate preliminary orbits and as starting points for a recursive procedure of corre- lation. We shall focus on optical observations of GEO showing some preliminary results testing the 2007 data by the ESA Optical Ground Station (OGS) at Teide Ob- servatory (Canary Islands), after an astrometric reductio n performed by University of Bern.
ABSTRACT BepiColombo (BC) is an ESA mission, including two satellites orbiting Mercury, to be lau... more ABSTRACT BepiColombo (BC) is an ESA mission, including two satellites orbiting Mercury, to be launched in 2015. JUNO is a NASA mission, already launched and on the way to Jupiter. Both S/C carry dedicated Ka - band transponders and other equipment to perform the most advanced interplanetary radio science experiments made possible by state of the art technology. A consistent set of models compatible with the accuracy of the range and range - rate measurements needs to include the dynamic model, the observation model, and the appropriate spacetime reference and coordinate systems. For BC the model for mercury - centric orbit has a limited need for relativistic terms, but the independent variable of the equations of motion needs to be a form of Dynamical Mercury Time, wit h comparatively large nonlinear differences with respect to the corresponding time coordinate with respect to the solar system barycenter (SSB). The dynamical model for the orbits of the planets Mercury and Earth need a complete PPN formulation and a large set of asteroids considered as perturbing masses; the very precise definition of the SSB is a challenge. The observation model requires as many as four separate iterations, plus a Shapiro term with second order corrections. For JUNO the dynamic model o f the jovicentric orbit requires direct and tidal perturbations from many satellites, and Jupiter relativistic terms including Lens - Thirring effect; the independent variable must be a Dynamical Jupiter Time. The observations model needs to include a Jupiter Shapiro effect, including an oblateness term.
Monthly Notices of the Royal Astronomical Society, 2014
Juno is a NASA mission launched in 2011 with the goal of studying Jupiter. The probe will arrive ... more Juno is a NASA mission launched in 2011 with the goal of studying Jupiter. The probe will arrive to the planet in 2016 and will be placed for one year in a polar higheccentric orbit to study the composition of the planet, the gravity and the magnetic field. The Italian Space Agency (ASI) provided the radio science instrument KaT (Ka-Band Translator) used for the gravity experiment, which has the goal of studying the Jupiter's deep structure by mapping the planet's gravity: such instrument takes advantage of synergies with a similar tool in development for BepiColombo, the ESA cornerstone mission to Mercury. The Celestial Mechanics Group of the University of Pisa, being part of the Juno Italian team, is developing an orbit determination and parameters estimation software for processing the real data independently from NASA software ODP. This paper has a twofold goal: first, to tell about the development of this software highlighting the models used, second, to perform a sensitivity analysis on the parameters of interest to the mission.
Proceedings of the International Astronomical Union, 2006
The distance between two confocal Keplerian orbits, called MOID in the literature, is a useful to... more The distance between two confocal Keplerian orbits, called MOID in the literature, is a useful tool to know if two celestial bodies can collide or undergo a very close approach. Two confocal orbits may get close at more than one pair of points, thus it is useful to compute not only the absolute minimum of the distance function d between
Monthly Notices of the Royal Astronomical Society, 2011
While building up a catalog of Earth orbiting objects, if the available optical observations are ... more 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 nongravitational 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.
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Papers by Giacomo Tommei