96386 - Stellar Dynamics

Learning outcomes

Theoretical understanding and working knowledge of the principal gravitational phenomena determining the structure, the dynamics and the evolution of stellar systems, from open and globular clusters, to galaxies, to galaxy clusters. At the end of the course, the student should be able to use in autonomy some of the advanced mathematical techniques needed in potential theory and in epicyclic theory.

Course contents

I] GENERALS

Introduction to the course. Gravitational field of point particles, superposition principle. Integral representation for generic distributions. The divergence operator's most important properties and its coordinate-free representation from the Gauss Theorem. Operational introduction to the one-dimensional and multidimensional Dirac Delta in Cartesian and Curved coordinates. Divergence of the field of generic distributions, Poisson equation. Integral proof of the First and Second Newton Theorem (homogeneous spherical shells), and from the Gauss Theorem. Coordinate-free representation of gradient, curl and laplacian operators. Exact fields and their property, potential and work. Closed fields. Stokes theorem, fields closed in domains simply and multiply connected. Existence of potential and its relation to the total energy of a particle. Potential difference as a line integral. Point mass potential. Potential of generic distributions, meaning of the additive constant. Poisson and Laplace equations. First and second identity of Green, uniqueness of the Poisson equation solution in finite volumes with assigned boundary conditions. Helmholtz Decomposition Theorem. Definition of concentric similar ellipsoids. Definition of homeoid. Third Newton theorem. Field inside heterogeneous ellipsoids. Co-area theorem. Confocal elliptical coordinates. Classification of the three families of associated quadrics. Elliptical coordinates: orthogonality, gradient, Laplacian. Application to the problem of ellipsoidal layer with internal null field. Heterogeneous ellipsoidal potential. Chandrasekhar Formula.

Introduction to the multipole expansion of potential in the far field. Monopole, dipole and quadrupole terms.

Introduction to the Green Function. Linear Differential Operators. Point potential as an explicit Green Function for Laplacian. Separation of variables in Cartesian coordinates. Fourier transforms in Rn. Green function in Cartesian coordinates. Green function in spherical coordinates, separation of variables. Rotational invariance and quantum azimuthal number m. Legendre trigonometric and algebraic differential equation. Singularities of ODEs, fixed and movable. Fuchs's theorem, regular, singular regulars and singular essentials points. Classification for Legendre equation. Frobenius method and quantum polar number. Legendre P and Q Functions. Legendre Polynomials. Rodrigues formulas, associated polynomial. Orthogonality of solutions with the Sturm-Liouville theory. Spherical harmonics. Cylindrical symmetry systems. Generating function for Legendre polynomials, moments of multipole. Polynomials of Gegenbauer. Addition theorem for spherical harmonics. Separation of variables for the Laplacian vacuum solution in cylindrical coordinates. Bessel equation and its properties: orthogonality of solutions, singular points. Asymptotic analysis of Bessel's functions for large values of the argument. Hankel's closure. Green function in cylindrical coordinates for Laplacian. Fourier-Bessel transforms. Infinitely thin, axisymmetric discs, potential in disc plane, homogeneous rings.

Thin disk rotation curve. Mestel's disc and exponential, implications for the dark matter halos. Potential of axisymmetric systems using elliptical integrals.

II] COLLISIONLESS SYSTEMS

Introduction to Epicyclic approximation. Curvilinear coordinates, velocity and acceleration in cylindrical coordinates. Newtonian equations in axysimmetric potentials, energy and Jz conservation. Eulero-Lagrange equations. Meridional plan and the effective potential. Equations of motion in the meridional plane, orbital families, circular orbits. Centrifugal barrier, zero velocity curves. Second order expansion of the effective potential. Vertical and radial epicyclic frequencies. Radial and vertical motion in the case of stable orbits, zero-velocity ellipses. Rayleigh criterion. First order angular motion, and determination of the axes for the epicyclic ellipse. Coulomb, harmonic and flat rotating curves: frequency and shape. Oort constants and the radial epyciclic frequency. Closed and rosette, orbits, pattern angular velocity, Lindblad kinetic waves.

Readings/Bibliography

The content of the course corresponds to an in-depth discussion of the the chapters (and a few selected exercises)

1, 2, 5, Appendix

of "Introduction to Stellar Dynamics" (L. Ciotti, Cambridge University Press). 

This is the reference textbook for the course, it contains all the informations needed for the preparation of the final examination.

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In case of particular interest in some of the treated arguments, additional suggested readings are 

"Dynamics of galaxies" (G. Bertin, Cambridge University Press)

"Galactic Dynamics" (J. Binney, S. Tremaine Princeton University Press)

"Galactic Astronomy" (J. Binney, M. Merrifield Princeton University Press)

"Dynamical evolution of globular clusters" (L. Spitzer Princeton University Press)

"Lecture Notes on Stellar Dynamics' (L. Ciotti Scuola Normale Superiore Pisa - Springer Verlag)

"Classical Dynamics" (Jose' & Saletan, Cambridge University Press)"Introduction to Stellar Dynamics" (L. Ciotti, Cambridge University Press)

Teaching methods

Class lectures, discussion of influential research papers on international journals

Assessment methods

Final oral examination. The examination is at most 45 minutes long, at the blackboard, organized in 3 sections (each 15 minutes long). In the first part the student illustrates the general concepts of a proposed subject (the aim is to verify the presentation abilities). In the second part the student is asked to solve a simple exercise (test of numerical abilities, and of a back-to-the-envelope estimate). In the third and last part a random argument from the program is discussed (test of the global preparation).

Teaching tools

Blackboard. Tools for online lectures. Lecture Notes.

Office hours

See the website of Luca Ciotti