### Timetable

### Description

Master’s Programme in Particle Physics and Astrophysical Sciences is responsible for the course.

Module where the course belongs to:

- PAP300 Advanced Studies in Particle Physics and Astrophysical Sciences

Optional for:- Study Track in Astrophysical Sciences

The course is available to students from other degree programmes.

Celestial Mechanics and Galaxies and Cosmology. The basic- and intermediate-level courses of Astronomy as well as Analytical Mechanics and FYMM I-II of Theoretical Physics are also recommended.

The course is primarily linked together with Small Bodies in the Solar System, and Galaxy Formation and Evolution.

The student understands the pros and cons with different integrators and is able to choose the best tool for the problem at hand. The student can derive the disturbing function for the problem at hand and is able to use the disturbing function to analytically study the long-term evolution of a dynamical system. The student is understands and is able to use tools to detect chaos in planetary systems. The student understands the principles of planetary migration and non-gravitational forces. The student understands the connection between the observed structure of the solar system and the dynamical effects that have shaped it since its formation.

The student will be able to calculate the relaxation and dynamical timescales for galaxies. The student will be able to calculate the gravitational potential for spherical

and flattened systems. The student will understand the basic principles of direct summation codes, tree codes and particle-mesh codes used to perform numerical

galaxy formation simulations. The student will be able to describe the orbits of stars in spherical, axisymmetric and simple non-axisymmetric potentials. The student will

understand the how the Boltzmann and Jeans equations can be used in galaxy dynamics. The student will be able to derive the tensor virial theorem. The student will

understand the basics of relaxation processes in galaxies and understand the thermodynamics of self-gravitating systems. The student will be able to derive the formula

for dynamical friction and understand its application. The student will understand the importance of galaxy mergers for galaxy evolution.

This course is lectured every two years, so the student should take it when it is on offer.

This course will be offered in the Spring semester, periods 3-4, every two years. The course will be lectured odd years, next time in the Autumn of 2019.

The planetary dynamics part starts with a short recap of the Celestial Mechanics course covering the 2-body problem, perturbation theory, and Lagrange's planetary equations. The N-body problem is formulated within the framework of Hamiltonian mechanics with special emphasis on integrals and constants of motion, integrable Hamiltonian systems, and the integrability of the N-body problem. Surfaces of section are outlined in integrable cases and foundation is laid for the study of dynamical chaos.

An essential part of astronomical dynamics deals with numerical integration. The student will build upon the knowledge gained during the Celestial Mechanics course and learn about more sophisticated numerical integration schemes such as symplectic integrators, the Bulirsch-Stoer algorithm, as well as the so-called MVS integrator. Particular attention is paid to the detection and characterization of chaos in planetary systems using, e.g., Lyapunov exponents.

We construct the disturbing function step by step and use it to develop an understanding for the most important secular and resonant perturbations in the solar system. We compare the analytical solution to the numerical results obtained with the integrators mentioned above.

We end the planetary dynamics part with a short introduction to planetary migration and non-gravitational forces, and discuss the implications of planetary dynamics on the structure and long-term evolution of the solar system.

The galactic dynamics part of the course introduces the field, which is an integral part of modern theoretical astrophysics. The course follows the outline of the second edition of the classic text "Galactic Dynamics" by Binney & Tremaine (2008).

We begin with a general overview of galaxies, their properties and classification followed by a discussion of relaxation and dynamical timescales. After this we discuss potential theory, how to compute the gravitational potential of galaxies and how to describe galaxies using spherical and flattened density distributions. Then orbit theory is discussed, specifically what kinds of orbits are possible in galaxies described by a spherically symmetric, or an axially symmetric potential.

We continue with a discussion of the equilibria of collisionless systems and the collisionless Boltzmann equation. We then introduce the Jeans and virial equations and with the help of them detect black holes and dark matter haloes in galaxies using observations of the kinematics of their stars. This is followed by a discussion on kinetic theory and the thermodynamics of self-gravitating systems. We end this part of the course with a discussion on dynamical friction and its applications and describe the related concepts of galaxy interactions and mergers.

The course will use chapters from these books:

- B. Gladman, J. Burns, Planetary Dynamics, 2011 (draft of a book under preparation).
- J. Binney, S. Tremaine, Galactic Dynamics, 2nd Ed., Princeton University Press, 2008.

Additional material:

- C. D. Murray & S. F. Dermott: Solar System Dynamics, Cambridge Univ Press, 1999.
- A. Morbidelli: Modern Celestial Mechanics: Aspects of Solar System Dynamics, Advances in Astronomy and Astrophysics, CRC Press, 2002.
- J. M. A. Danby: Fundamentals of Celestial Mechanics, 2nd Ed., Willmann-Bell, Inc., 1992.
- Sparke & Gallagher: Galaxies in the Universe, 2nd Ed., Cambridge Univ Press, 2007.
- Bertin: Dynamics of Galaxies, Cambridge Univ Press, 2000.
- Binney & Merrifield: Galactic Astronomy, Princeton Univ Press, 1998.

Two hours of weekly lectures and written problem sets. The problem set session will led by the course assistant and there the correct solutions will be discussed and presented.

In addition the course requires a supervised course project related to orbit integration.

To pass with a grade 1/5 requires 43.3% of the maximum exam points, for the highest grade 5/5 the requirement is 86.7% of the maximum exam points. The maximum points from the final exam is 30 and an additional 6 points can be acquired from the problem sets. Additional points are only awarded for problem set points that exceed the minimum level of one third, which isrequired in order to have the right to take part in the final exam.

The course is composed of partly compulsory exercises, a project, and a final exam. In the final exam, the student can have specific lecture material with him/her.