03391 - Celestial Mechanics

Learning outcomes

At the end of the course, students will gain an understanding of the key methods of celestial mechanics and some of its most important astrophysical applications. They will become familiar with the main results concerning the gravitational N-body problem with a particular emphasis on the two- and three-body problem. Additionally, students will acquire a basic knowledge of perturbation methods and numerical integration techniques for orbital motion.

Course contents

The course provides an overview of the fundamental principles of Celestial Mechanics, with an emphasis on both theoretical aspects and astrophysical applications. The main topics covered include:

1. Introduction and theoretical foundations

• Historical overview of Celestial Mechanics

• Review of classical mechanics: Newtonian, Lagrangian, and Hamiltonian formulations

• The gravitational N-body problem

• Integrals of motion and conservation laws

2. The two-body gravitational problem

• Reduction to the motion of a single particle in a central force field

• Reduced mass

• Elliptical, parabolic, and hyperbolic orbits

• Kepler’s laws

• Coordinate systems and orbital elements

• Kepler’s equation

• Action-angle variables

• Orbital maneuvers

3. The three-body problem

• The gravitational three-body problem

• The circular restricted three-body problem

• Jacobi integral

• Tisserand’s criterion

• Zero-velocity surfaces

• Lagrangian equilibrium points

• Linear stability of Lagrangian points

• Orbits around Lagrangian points

4. Perturbation theory and applications

• Introduction to perturbation theory

• The disturbing function

• Perihelion precession of Mercury: classical and relativistic treatment

5. Numerical methods

• Stability of a numerical integrator for ordinary differential equations

• Numerical integration of orbits

• Introduction to symplectic integrators

Readings/Bibliography

Required materials:

• Lecture notes and slides provided by the lecturer.

Recommended textbooks:

• S. Tremaine, "Dynamics of Planetary Systems", Princeton University Press (selected parts of Chapters 1, 2, 3, 4, and 5).

• M. Valtonen & H. Karttunen, "The Three-Body Problem", Cambridge University Press (Chapter 5).

Further reading (for optional in-depth study):

• A.E. Roy, "Orbital Motion", Taylor & Francis (selected parts of Chapters 2, 5, and 7).

• C.D. Murray & S.F. Dermott, "Solar System Dynamics", Cambridge University Press (selected parts of Chapters 1, 2, 3, 6, and 9).

• L.D. Landau & E.M. Lifshitz, "Mechanics", Butterworth–Heinemann (selected parts of Chapters 1, 2, 3, and 7).

• H. Goldstein, C. Poole, J. Safko, "Classical Mechanics" (3rd edition), Addison-Wesley (selected parts of Chapters 1, 2, 3, 8, 9, and 10).

Teaching methods

The course will be delivered through in-person lectures in the classroom, supported by the use of a blackboard and a video-projector. The main theoretical concepts will be developed progressively, following a deductive and guided approach. Lectures will also include time dedicated to the analysis and the resolution of applied problems, in order to promote active learning and consolidate the theoretical knowledge acquired.

Assessment methods

Learning will be assessed through an oral examination, during which three distinct topics covered in the course will be discussed. The first question will be based on a topic freely chosen by the student. The exam may also include the resolution of problems similar to those discussed in class. The aim of the oral exam is to evaluate the student’s understanding of both the methods of celestial mechanics and their astrophysical applications.

Grading criteria (final mark out of 30):

• 18–19: knowledge limited to a small number of course topics; analytical skills evident only with guidance from the instructor; generally correct use of language.

• 20–24: sufficient knowledge of a fair number of topics; independent analysis limited to routine problems; correct use of language.

• 25–29: solid knowledge of a wide range of topics; ability to make autonomous and critical assessments; good command of technical terminology.

• 30–30L: comprehensive understanding of course topics; fully autonomous critical thinking and connections across topics; full command of specialized terminology and strong argumentative and reflective skills.

Students with learning disabilities or temporary or permanent disabilities: please contact the relevant University office promptly (https://site.unibo.it/studenti-con-disabilita-e-dsa/eng). The office will advise students of possible adjustments, that will be submitted to the professor for approval 15 days in advance. He/she will evaluate their suitability also in relation to the academic objectives of the course.

Teaching tools

The following tools will be used to support teaching activities:

• Blackboard and video-projector for the presentation and development of theoretical and applied content during lectures.

• Lecture slides/notes provided by the instructor (in digital format), available on the Virtuale learning platform.

• Computational software (in Fortran90) and visualization tools (in Python) for the numerical integration and graphical representation of orbits.

Office hours

See the website of Federico Marinacci