Risorse bibliografiche
Risorsa bibliografica obbligatoria
Risorsa bibliografica facoltativa
Scheda Riassuntiva
Anno Accademico 2019/2020
Scuola Scuola di Ingegneria Industriale e dell'Informazione
Insegnamento 083794 - ORBITAL MECHANICS
Docente Colombo Camilla
Cfu 10.00 Tipo insegnamento Monodisciplinare

Corso di Studi Codice Piano di Studio preventivamente approvato Da (compreso) A (escluso) Insegnamento

Obiettivi dell'insegnamento

Course overview

The course gives complete notions on the orbital mechanics of a spacecraft, which are required in the process of designing a space mission. It gives the knowledge to design the space trajectory and control it and the basis of performing the mission analysis.

Theoretical approaches and numerical techniques will be explained to perform the trajectory design of terrestrial or planet-centred and interplanetary missions.

The course leads on from the restricted two-body problem and introduces to the characterisation, design and maintenance of planet-centred orbits in presence of orbit perturbations. Transfer manoeuvres are explained for planet-centred orbit transfer, escape and capture phases of the mission. The module continues by investigating various techniques for interplanetary trajectory design, gravity assists manoeuvres, rendezvous and docking phases. Preliminary concepts for optimisation of interplanetary impulsive trajectories with deep space manoeuvres will be introduced. Finally, the course will introduce the circular and elliptical three body problem and introduce to the use of dynamical system theory for transfer in the Earth-Sun system. Two lectures delivered in collaboration with external experts will show the practical solution of part of the mission analysis of a specific space missions as performed in studies for the European Space Agency.

Course aims

- To provide students with the techniques for the design of spacecraft trajectories around a planet or for interplanetary missions

- To provide students with an appreciation and understanding of the fundamental dynamics behind spaceflight

- To provide an overview of the techniques for trajectory optimisation and control and orbit maintenance

Risultati di apprendimento attesi

- To select the orbit for a planet-centred mission

- To implement in a computer code trajectory design techniques and characterisation of planetary-centred orbit

- To use techniques for numerical and semi-analytical modelling of perturbations for the characterisation of a new spacecraft orbit

- To design and optimise the manoeuvre sequences to perform and interplanetary mission with impulsive and gravity assists manoeuvres

- To use its own creativity to develop ideas for the solution of an interplanetary trajectory optimisation problem

- To implement the theory into an efficient and clear computer code for mission analysis

- To develop skills for critical review of its own results obtained via numerical exercises or computer simulation

- To actively participate to numerical and informatics classes for the verification of models and techniques related to space flight mechanics

Soft skills:

- To develop a group project ad organise the team work.

- To develop presentation skills during the final presentation of the lab project

- To draft a report of a trajectory design project

Argomenti trattati

Reference systems in space: astronomy concepts; celestial sphere; rectangular and spherical coordinates: local equatorial and ecliptic; Earth movements; the Solar system; time measurement; coordinates transformations; spherical trigonometry; interplanetary navigation; ground tracks.

Keplerian orbits: the fundamental problem of celestial mechanics; the Keplerian model; restricted two-body problem; Kepler equation; position and velocity as function of time; ephemerides; angular momentum; energy; orbital elements and state vector; ground tracks.

Impulsive dynamics: fundamentals; variable mass systems dynamics; mission launcher; multi-stage launcher; design and trajectory optimisation.

Orbital manoeuvres: thrust in vacuum; review of Hohmann transfer, bi-elliptical transfer, out-of-plane manoeuvre; station keeping; orbital transfer; rendezvous and docking.

Gravitational motion and interplanetary trajectories: escape velocity; orbit insertion; gravity assist manoeuvres; interplanetary trip; launch window optimisation; sphere of influence; patched conics; Tisserand plane; proximity manoeuvring.

Atmospheric trajectories: space vehicle manoeuvring in launch and entry atmospheric phases.

Orbit perturbation modelling and applications: the perturbation problem; Cowell, Encke, variational approaches; Lagrange and Gauss equations; disturbing function; semi-analytical techniques for long-term orbit propagation; solar radiation pressure, third-body perturbation, atmospheric drag, the Earth gravity potential model; gravity harmonics; design of frozen orbits.

Restricted three-body problem: N-body problem, Three-body problem, restricted-three body problem, Lagrange points, Jacobi constant, periodic orbits, introduction to manifold dynamics for missions at Libration point orbits.


Skills in programming with Matlab (preferably) or Python or C.

Modalità di valutazione

The course includes frontal lessons (with the aid of both blackboard and slides) and labs to face numerical examples on theory topics. In addition, computer labs and tutorials will show the application of the numerical techniques to spacecraft trajectory design.

The exam is divided in three parts to consolidate the theoretical explanation and the practical application of them.

Written exam with exercises: to test and consolidate the solution of practical problems and the application of the theory to find solutions to them.

Individual oral discussion (in English): on the knowledge of the theory to be held as soon as the written exam turns out to be positive.

Group project on the design of the trajectory for a mission: group presentation and short report of the mission design computer assignment.

Risorsa bibliografica obbligatoriaH. Curtis, Orbital Mechanics for Engineering Students, Editore: Butterworth-Heinemann, Anno edizione: 2010, ISBN: 978-0080977478

2009 (2nd Edition) or 2010 (3rd Edition)

Risorsa bibliografica facoltativaV. Chobotov, Orbital Mechanics, Editore: AIAA Education, Anno edizione: 2002, ISBN: 978-1563475375

3rd Edition

Risorsa bibliografica facoltativaD. A. Vallado, Fundamentals of Astrodynamics and Applications, Editore: Springer (Space Technology Library), Anno edizione: 2007, ISBN: 978-0387718316
Risorsa bibliografica facoltativaR. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, Editore: Revised Edition, Aiaa Educational Series, Reston, Anno edizione: 1999
Risorsa bibliografica facoltativaM.H. Kaplan, Modern Spacecraft Dynamics and Control , Editore: Ed. Wiley & Sons
Risorsa bibliografica facoltativaK.J. Ball, G.F. Osborne, Space Vehicle Dynamics, Editore: Clarendon Press

Forme didattiche
Tipo Forma Didattica Ore di attività svolte in aula
Ore di studio autonome
Laboratorio Informatico
Laboratorio Sperimentale
Laboratorio Di Progetto
Totale 100:00 150:00

Informazioni in lingua inglese a supporto dell'internazionalizzazione
Insegnamento erogato in lingua Inglese
Disponibilità di materiale didattico/slides in lingua inglese
Disponibilità di libri di testo/bibliografia in lingua inglese
Possibilità di sostenere l'esame in lingua inglese
Disponibilità di supporto didattico in lingua inglese
schedaincarico v. 1.6.1 / 1.6.1
Area Servizi ICT