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, principle of superposition. Integral representation for any distributions. Most important properties of the divergence operator and its coordinate-free representation starting from Gauss's Theorem. Operational introduction to the one-dimensional and multidimensional Dirac Delta in Cartesian and curvilinear coordinates. Calculation of the divergence of the field of extended distributions, Poisson's equation for the field. Direct proof of the First and Second Newton's theorem (homogeneous spherical shells). Alternative demonstration using Gauss's theorem. Coordinate-free representation of the gradient, curl, and Laplacian operators. Notes on differential forms. Exact fields and their properties, potential and work. Closed fields. Stokes' theorem, closed fields in simply and non-simply connected domains. Existence of the potential and its connection with the total energy of a particle. Potential difference as a line integral. Formal calculation of the potential of a point mass. Potential of extended distributions, general expression and discussion of the meaning of the additive constant. Poisson and Laplace equations. First and second Green's identities, uniqueness of the solution of the Poisson equation in bounded volumes with prescribed boundary conditions. Field inside cavities with equipotential boundary. Helmholtz Decomposition Theorem. Definition of concentric and similar ellipsoids. Definition of homoeoid. Statement of the Third Newton's Theorem for finite homoeoids. Field inside a heterogeneous hollow homoeoid from the principle of superposition. Co-area theorem, relationship with the field of homoeoids. Definition of confocal ellipsoidal coordinates. Classification of the three families of associated quadrics. Ellipsoidal coordinates: orthogonality, gradient, Laplacian. Application to the problem of the ellipsoidal layer with zero internal field. Potential of the heterogeneous ellipsoid. Chandrasekhar's formula.

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

Introduction to the concept of Green's function for linear differential operators and their use in solving nonhomogeneous problems. The potential of a material point as an explicit example of a Green's function for the Laplacian. Separation of variables for the Laplacian in Cartesian coordinates. Fourier transform and inverse transform in Rn, the case of the Dirac Delta. Green's function in Cartesian coordinates. Green's function in spherical coordinates. Separation of variables. Rotational invariance and the azimuthal quantum number m. Orthogonality of azimuthal functions. Associated Legendre equation for the latitude angle, transformation into an algebraic equation. Outline with examples of singularities of ODEs, both mobile and fixed. Fuchs' theorem, regular points, regular singularities, and essential singularities. Classification for the Legendre equation. Frobenius method and polar quantum number. Legendre functions and associated functions P and Q. Legendre polynomials. Rodrigues formulas, norm of associated polynomials. Orthogonality of solutions with Sturm-Liouville theory. Spherical harmonics as eigenfunctions of the angular part of the Laplacian. Systems with cylindrical symmetry. Generating function for Legendre polynomials, multipole moments. Gegenbauer polynomials. Addition theorem for spherical harmonics. Separation of variables for the vacuum solution of the Laplacian in cylindrical coordinates. Bessel equation and its properties: orthogonality of solutions, singular points. Asymptotic analysis of Bessel functions for large values of the argument. Closure relation and Hankel transform. Green's function in cylindrical coordinates for the Laplacian. Any density potential with Fourier-Bessel transforms. Case of axisymmetric systems. Infinitely thin axisymmetric disks, potential in the plane of the disk, 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 the epicyclic approximation. Notes on curvilinear coordinates, velocity and acceleration in cylindrical coordinates. Newtonian equations of motion in general axisymmetric potentials, conservation of energy and Jz. Deduction of equations from the Euler-Lagrange equations. The meridional plane, its motion, and effective potential. Equations of motion in the meridional plane, orbital families, circular orbits and their (equivalent) equations. Interpretation of total energy as energy for motion in the meridional plane, extremum properties for the energy of circular orbits, centrifugal barrier, zero-velocity curves. Development of the effective potential to second order. Frequency of vertical and radial epicycles. Radial and vertical motion on the epicycle in the case of stable orbits, zero-velocity ellipses. Rayleigh criterion and examples of applications. First-order angular motion, coordinates on the equatorial plane referred to the deferent, equation of the epicycle on the equatorial plane, and determination of the axes for the epicyclic ellipse. Epicycles in Coulomb, harmonic, and flat rotation potentials: frequency and shape. Relation of Oort constants to the radial epicyclic frequency. Closed, rosette, and open orbits: closure conditions, pattern angular velocity, Lindblad kinetic waves, and the dynamical phenomenology of disks.

Readings/Bibliography

The course consists of the presentation, discussion, and in-depth study of the chapters in class.

Chapters 1, 2, 5 of "Introduction to Stellar Dynamics" (L. Ciotti, Cambridge University Press, 2021), and some selected exercises.

The indicated chapters, the explanations in class, and the Appendices of the book contain ALL the necessary and sufficient information for the complete preparation of the exam. No additional material is required.

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In case of personal interest, for further in-depth study (not required for the exam) of the topics covered in class, the following texts are indicated as particularly comprehensive:

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

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

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

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

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

6. "Classical Dynamics" (Jose' & Saletan, Cambridge University Press)

Teaching methods

Lectures include discussion and a detailed, comprehensive derivation on the board of all formulas and concepts presented. Students who have attended the classroom course and use the above-mentioned course materials (assuming they have the physical and mathematical skills required for a relatively advanced course) are fully prepared to take the exam.

Assessment methods

Oral exam, on the blackboard, in the classroom. The exam has a maximum duration of 45 minutes and is divided into three parts (each lasting up to 15 minutes). Free-form topics are not permitted.

First part: Students are asked to present the general concepts of a topic from the syllabus (the purpose is to test their ability to present and analyze in depth, and their understanding of the topic in its astrophysical context).

Second part: Students are asked to solve and discuss a simple problem (the purpose is to test formal/numerical skills and the ability to make order-of-magnitude estimates of physical phenomena).

Third and final part: A question on a course topic unrelated to the first and second questions, requiring a simple answer (e.g., a given formula, a given result) without in-depth discussion (the purpose is to test overall preparation).

Each of the three questions is immediately scored numerically from 0 to 10:

1) Seriously incomplete/incorrect answer: less than 6

2) The knowledge demonstrated, even if incomplete, reveals sufficient preparation to allow the result to be used in other courses or, in any case, does not impede necessary in-depth study: between 6 and 8

3) Complete and comprehensive answer demonstrating full mastery of the topic and independent reflection on the discussion: between 8 and 10.

The final exam grade is the sum of the three grades. Please note that 

• The order of presentation, clarity of language, and ability to formalize the mathematical aspects in a technically acceptable manner are all factors that influence the grade for each question.
• Pursuant to the CdS resolution of June 20, 2025, a successfully passed exam grade cannot be rejected more than twice; In case of retaking the exam, the final grade will be that of the last exam taken.
• Students with learning disabilities or temporary or permanent disabilities: contact the office https://site.unibo.it/studenti-con-disabilita-e-dsa/it

 

Teaching tools

Classroom whiteboard. Online teaching support available if needed. Student reception.

Office hours

See the website of Luca Ciotti