The restricted three-body problem (RTBP) is the ideal model to design unique solutions, ranging from Lagrange point orbits to low energy transfers. These orbits embed the effect of two gravitational attractions in a natural way, and therefore they are more accurate than the conics, solutions of the classic two-body problem. However, when three-body orbits are reproduced in the real solar system model, large errors are found. That is, as the three-body orbits are defined in the regions of phase space where the sensitivity is high, the additional terms of the real solar system model produce large effects along the orbits. The core of this thesis is to present an automatic algorithm for the correction of orbits in the real solar system. The differential equations governing the dynamics of a massless particle are written as perturbation of the RTBP in a nonuniformly rotating and pulsating frame by using a Lagrangian formalism. The refinement is carried out by means of a multiple shooting technique, and the problem is solved for a finite set of variables. The generality of the algorithm lies in the possibility of handling both constrained and unconstrained boundary conditions. In the latter case, the problem is solved by minimising a certain performance index. Once the problem is stated, the gradient of the objective function, as well as the Jacobian of the constraints are computed and assembled in an automatic fashion. Results are given for the dynamical substitutes of the collinear points of several three-body systems. Periodic and quasi-periodic orbits in the framework of the RTBP (e.g., halo orbits) are refined in the full gravitational solar system model by means of the proposed method. The trajectory-refinement algorithm has been implemented with the idea of being versatile. With minor adjustments to the code backbone it can be applied to a large variety of practical astrodynamics problems, from stable orbits to optimised propelled trajectories and orbits that exploit the intrinsic dynamics of the solar system.
Il problema ristretto dei tre corpi (RTBP) è il modello ideale per calcolare soluzioni uniche, che vanno da orbite nell’intorno dei punti Lagrangiani ai trasferimenti a basso consumo energetico. Queste traiettorie incorporano l’effetto di due attrattori gravitazionali in modo naturale, e si prestano dunque ad uno studio più accurato rispetto alle coniche, soluzioni del classico problema di Keplero. Tuttavia, quando le orbite dei tre corpi sono riprodotte nel modello reale del sistema solare, si trovano grandi errori. In particolar modo, quando queste orbite sono definite nelle regioni dello spazio delle fasi in cui la sensibilità è elevata, le perturbazioni dovute ad altri corpi producono grandi effetti lungo le orbite. Il fulcro di questo lavoro di tesi giace nello sviluppo di un algoritmo automatico per la correzione delle orbite al modello gravitazionale completo che descrive il sistema solare. Le equazioni differenziali che regolano la dinamica di una particella priva di massa sono scritte come perturbazione del RTBP in un sistema non uniformemente rotante e pulsante, utilizzando un formalismo Lagrangiano. La procedura di affinamento avviene mediante una tecnica di multiple shooting, e il problema è risolto per un insieme finito di variabili. La generalità dell’algoritmo consiste nella possibilità di gestire condizioni al contorno sia vincolate che non vincolate. In quest’ultimo caso, il problema è risolto tramite minimizzazione di un determinato indice. Formulato il problema, il gradiente della funzione obiettivo e il Jacobiano dei vincoli vengono calcolati e assemblati in modo automatico. Vengono forniti i risultati per i sostituti dinamici dei punti collineari di diversi sistemi a tre corpi. Orbite periodiche e quasi-periodiche nel quadro del RTBP (ad esempio orbite halo) sono affinate nel modello completo a n corpi mediante il metodo proposto. L’algoritmo di affinamento della traiettoria è stato implementato con l’idea di essere versatile. Con aggiustamenti minori al codice, esso può essere applicato ad una grande varietà di problemi pratici dell’Astrodinamica: da orbite stabili ad ottimizzazione di traiettorie propulse e orbite che sfruttano le dinamiche intrinseche del sistema solare.
Automated trajectory refinement of three body orbits in the real solar system model. Dynamical substitutes of Lagrangian points and quasi-periodic orbits about them
DEI TOS, DIOGENE ALESSANDRO
2013/2014
Abstract
The restricted three-body problem (RTBP) is the ideal model to design unique solutions, ranging from Lagrange point orbits to low energy transfers. These orbits embed the effect of two gravitational attractions in a natural way, and therefore they are more accurate than the conics, solutions of the classic two-body problem. However, when three-body orbits are reproduced in the real solar system model, large errors are found. That is, as the three-body orbits are defined in the regions of phase space where the sensitivity is high, the additional terms of the real solar system model produce large effects along the orbits. The core of this thesis is to present an automatic algorithm for the correction of orbits in the real solar system. The differential equations governing the dynamics of a massless particle are written as perturbation of the RTBP in a nonuniformly rotating and pulsating frame by using a Lagrangian formalism. The refinement is carried out by means of a multiple shooting technique, and the problem is solved for a finite set of variables. The generality of the algorithm lies in the possibility of handling both constrained and unconstrained boundary conditions. In the latter case, the problem is solved by minimising a certain performance index. Once the problem is stated, the gradient of the objective function, as well as the Jacobian of the constraints are computed and assembled in an automatic fashion. Results are given for the dynamical substitutes of the collinear points of several three-body systems. Periodic and quasi-periodic orbits in the framework of the RTBP (e.g., halo orbits) are refined in the full gravitational solar system model by means of the proposed method. The trajectory-refinement algorithm has been implemented with the idea of being versatile. With minor adjustments to the code backbone it can be applied to a large variety of practical astrodynamics problems, from stable orbits to optimised propelled trajectories and orbits that exploit the intrinsic dynamics of the solar system.File | Dimensione | Formato | |
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https://hdl.handle.net/10589/93675