Gravitational Dynamics 


John Scalo

Office: R.L. Moore 17.220
Phone: 471-6446 (office), or 478-2748 (home)

Description and Tentative Syllabus

"Gravitational Dynamics" is a course that has never been taught before, so this syllabus is tentative and subject to some change as we work out the kinks in what I have planned. The faculty decided in 1999, as part of the revision in the graduate program, that such a course needed to be introduced into our curriculum. The course was meant to coherently consolidate a large number of theoretical topics involving gravitational phenomena that were not being covered, or only marginally covered, by other existing courses and yet were regarded as essential background knowledge for working astronomers. The range of topics is, however, so broad that no single faculty member was equipped to teach the course. For this reason the course could not be offered until a faculty member studied the literature on these topics and designed a syllabus and preliminary notes. I volunteered (I wanted to learn about this stuff) and was granted a Course Development Leave during the Spring 2000 semester for this purpose.
There were two main problems I encountered in developing the materials. First, the number of topics and important applications is extremely large, since gravity plays a role in most phenomena in astrophysics, from the solar system to cosmology. Examples include the chaotic evolution and stability of the solar system, the evolution of interacting binary stars, the collective dynamics of stars in clusters and galaxies, the role of gravitational instability in star formation, general relativistic phenomena associated with black holes and gravitational radiation, and many more. How can such a range of topics be covered adequately in a single semester? The second problem involves the varied backgrounds of students enrolled in the course, involving different exposures to math/physics/astronomy topics. The description below represents my attempt to resolve these two problems.

My primary goal is to give the students a level of familiarity and comfort with all the concepts, issues, and techniques involved in a broad range of astrophysical problems. After taking this course, students should be able to read the literature in any of these fields and not feel completely lost because of lack of background. This is not a course to train "specialists" in "gravitational dynamics", but a course that provides a backbone for future research and teaching. Similarly, the objectives of the course are not be aimed at preparation for other courses, especially since only a fraction of students will enroll in these other courses.
These principles imply that most topics must be covered at some "middle level." Comprehensiveness, long technical derivations, and advanced topics must, in most cases, be sacrificed for a clear exposition of the basic physical principles and applications involved. I especially want students to become acquainted with a large number of applications throughout diverse astronomical fields, and this means that considerable time will be spent on order of magnitude, rough and dirty, back-of-the-envelope calculations. Developing skills with these approaches is, in my view, more useful than being able to reproduce long and technical derivations. This lack of rigor will be partly compensated in at least two ways. First, many of the homework problems will consist of asking students to fill in the parts of derivations that are skipped in lectures. This procedure has the advantage of forcing students to keep up with the course notes. Second, an extensive set of recent journal references will be provided that should provide the student with adequate resources to investigate in more detail topics of relevance to their specific dissertation research or interests. (For starters, about 50 recent or seminal papers covering a broad range of topics have [so far] been downloaded electronically and stored on 90Mb Zip disks as pdf files. Copies of these disks can be made available to students or the contents put on a web page for easy access by students.)


There will be no exams, but there will be several types of homework assignments.
a. The usual "problem sets". These will partly consist of filling in derivations skipped in class, as noted above, but also other problems that I think will be of benefit and keep you thinking about the material. There will also be occasional assignments that ask you to search for relevant papers on a given topic and turn in a list of references and/or 1st page of each article.
b. Two more extended problems that you can work on over the first and second halves of the semester. I will explain what these are in class, but they will be problems that have not been solved in the literature. There is the possibility that we will end up with publishable results here. You will have some flexibility in the approach you take.
c. Students will be assigned to write one "review paper" demonstrating a thorough investigation of a chosen topic. A wide variety of topics will be supplied (see syllabus below), and you can add more-the goal is to get you to work in depth on a topic of direct interest to you. This is a flexible project; short reviews on two or more topics will be acceptable also. A short presentation of the "state of the art" in your chosen topic will probably be required at the end of the semester, depending on time constraints.

Grading: 50% from homework problem sets (probably assigned every 1.5 weeks on average), 20% from participation in the two extended homework problems, and 30% from the in-depth "review paper" on topic of student's choice.

There is no single text that covers most of the topics in this course. Two major books that cover many of the topics in this course are: (a) Solar System Dynamics, by C. D. Murray and S. F. Dermott (2000) for dynamics of small numbers of objects and most solar system applications; and especially (b) Galactic Dynamics, by J. Binney and S. Tremaine (1997) for most everything else. Unfortunately both of these texts are too advanced (for most students and for me too) and too comprehensive to be of use as primary textbooks except possibly as supplements and organizational resources. Instead, a number of exerpts and chapters from various texts (including the two mentioned above), as well as some choice journal papers, will be made available in Peridier Library.

The basic order of topics generally proceeds from smaller to larger numbers of interacting objects, then returning to general relativistic topics at the end. I will only list the main sections and major topics. In each case "Application" refers to an astronomical problem of current interest that we will try to examine in some detail. I have tried to keep a balance between the solar system, stellar, and galactic scales in the choice of topics and especially the applications. These applications can of course be varied and are only meant to be illustrative. Students will be encouraged to suggest topics to be covered that I may not have thought of. (A larger list is given below.)

GRAVITATIONAL DYNAMICS: Topics and Tentative Syllabus

  • Review of Vectors and Dynamics
    Arithmetic of, calculus of, dot, cross, div, curl, Laplacian; div. theorem; Gauss' theorem.
    Polar, cylindrical, and spherical coordinates.
    Velocity, acceleration, angular momentum in rect., polar, spherical systems.
    Newton's laws for particle, system of particles
    Angular momentum, energy

  • Gravity
    Properties of Conservative forces
    Potential energy
    Poisson equation
    Virial Theorem
    Calculation of gravitational force and potential of objects
    Point mass, line, annulus, disk
    Spherical shell, homogeneous sphere
    More challenging configurations
    Laplace equation
    Potential-density pairs
    Application: Rotation curves of galaxies and dark matter

  • Orbits
    Two-body problem
    Gravitational encounters
    Two-body scattering and approximate relaxation time
    Dynamical friction (M>>m)
    Impulse approximation (rapid encounters)
    Orbits in a smooth potential
    Epicyclic motion (nearly circular orbits)
    Harmonic motion
    Orbits in spherical and axisymmetric potentials
    Corotation, Lindblad resonances (planar, non-axisymmetric potentials)
    Box, tube, shell orbits
    Orbits in a bar potential
    3-body problem (schematic)
    Application: Chaotic evolution of planetary systems

  • Interlude: Order of Magnitude Estimates
    Large numbers of applications done quick and dirty. E.g. stellar pulsation peiod-density relation, temperature distribution for simple accretion disk model, gravitational focussing in accretion of a moving sphere, Schwarzschild radius of a black hole

  • Tides
    Tidal forces-simple derivation; Roche limit
    Expansion of the potential
    Spin-orbit coupling
    *Close binary stars
    Applications: tidal tails; binary formation by tidal capture; galactic tides and mass extinctions; Europa's subsurface ocean; warps and gaps in disks; stellar tidal disruption by supermassive black holes

  • *Disk Dynamics and Spiral Waves-from protostars to galaxies
    Winding Problem
    Waves: dispersion relation, propagation, damping
    Density wave theory
    Spiral arm generation
    Bars and warps
    Stability of collisionless disks: Toomre condition

  • Gravitational Kinetics
    Liouville, Boltzmann, Vlasov
    Moment (fluid) equations: Virial theorem (again); Jeans' theorem
    Collisionless systems: models for spherical and elliptical galaxies
    Fokker-Planck equation: diffusion coefficients and solutions
    N-body approach: techniques, special hardware, advantages, disadvantages. Recent advances center on hybrid and hierarchical N-body algorithms (e.g. NBODY, KIRA, TREE, P--3M) and high-speed special-purpose computational hardware (e.g. GRAPE, FGPA processors).
    Application: Sinking satellites; heating of galactic (and other) disks (more applications below)

  • *Equilibrium Structures
    Spherical systems: polytropic models of stars, clusters, interstellar clouds
    The isothermal sphere.
    Disk equilibria-Mestel disk; vertical structure
    Triaxial systems (brief)
    Approach to equilibrium: phase mixing, violent relaxation

  • Star and Galaxy Clusters
    Virial theorem
    Relaxation time for various types of systems
    Mass segregation; evaporation and ejection
    Tidal disruption; encounter with interstellar clouds
    Tidal disk shocking
    Core collapse
    Hard binaries; influence of central black hole
    Application: Detailed evolution of a globular cluster

  • Self-gravitating Fluids
    Conservation equations
    Perturbation analysis
    Gravitational instability
    Jeans length; application to early universe, ISM, turbulent medium
    Toomre condition for axisymmetric stability of rotating disk
    Formulation with boundary conditions: disk example; filament
    Fragmentation--relevance for giant planets, binaries, star and galaxy formation
    Numerical simulations of self-gravitating fluids: interstellar clouds and star formation

  • *General Relativy (baby level)
    Metric, curved spacetime, energy-momentum tensor, field equations.
    GR effects near compact objects
    Black holes
    Gravitational radiation: theory, astronomical sources, and future observatories
    Gravitational lensing.

SOME APPLICATIONS AND REFERENCES (students are asked to update parts of this as part of their class work)

  • Planetary origins: planetesimal N-body dynamics (Richardson, D. C. et al. 2000, Icarus, 143, 45).
  • Formation of the Oort Cloud (Duncan et al 1987 AJ, 94, 1330; Levison et al. 1999)
  • Uranus and Neptune as scattered bodies (Thommes, Duncan, Levison 1999)
  • Giant extrasolar planets at small stellar distances/large eccentricities: scattering? (Weidenschilling & Marzari 96, Nature, 384, 619; Lin et al. 1996, Nature, 380, 606; Armitage & Hanson 1999, Nature, 402, 633)
  • Chaos in the Solar System (Wisdom 1987, Icarus, 72, 241; Duncan & Quinn 1993 ARAA, 31, 265);
  • Planetary orbits (Laskar 1989, 1990, Sussman & Wisdom 1992, Science, 257, 56);
  • Obliquity of Mars (Touma & Wisdom 1993, Science, 259, 1294);
  • Rotation of Hyperion (Wisdom et al. 1984, Icarus, 58, 137)
  • Runaway OB stars (Leonard & Duncan 1988, AJ, 96, 222; 1990, AJ, 99, 608)
  • Binary formation by tidal capture (Kim & Lee 1999, A&A, 347, 123)
  • Tidal evolution of binary systems (see Witte & Savonije 1999 A&A, 350, 129)
  • Coalescing binaries (Ho&Lai 1999 MN, 308, 153; Lee et al 1999 MN, 308, 780)
  • Binary-single-star scattering: series of papers by Hut et al; see Heggie, D.C. et al. 1996, ApJ, 467, 359 for paper VII.
  • Formation of hard binaries and cluster evolution ( )
  • Star clusters with massive central black holes (Duncan & Shapiro 1982, ApJ, 253, 921, Paper IV)
  • Evolution of dense clusters (Quinlan & Shapiro 1989, ApJ, 343, 725; 1990, ApJ 356, 483; Spurzen, R. & Gieisz, M. 1996, MN, 283, 805; Spurzen, R. & Aarseth, S. J. 1996, MN, 282, 19; Aarseth & Heggie, D.C. 1998, MNRAS, 297, 794)
  • Tidal shocking of globular cluster system (Gnedin et al. 1999 ApJ, 514, 109; 522, 935)
  • Formation of dwarf galaxies and brown dwarfs in tidal tails (Mendesde Oliveira et al. 2000 for refs; also Lin et al. 1998 Science for brown dwarfs)
  • Galactic tides and mass extinction by comets (Matese & Whitmire 1996, ApJ Lett., 472, L41; 1998 ??)
  • High-velocity clouds as tidal stripping from Local Group Dwarfs (Mallouris et al. 1999/00)
  • Subsurface ocean on Europa and tidal heating (Carr et al. 1998 Nature, 391, 363; 1999 DPS vol.31 Moore & Schubert)
  • Star formation in starburst/ultraluminous galaxies by encounters (see Genzel et al. 1998, Nature, 395, 859; enhanced nuclear accretion: ____; tidal compression (Das&Jog 1999 ApJ, 527, 600).
  • Tidal disruption (and tidal disruption flares) of stars by massive central black holes (Magorrian & Tremaine 1999 MN, 309, 447; Ulmer 1999 ApJ, 514, 180)
  • Tidally induced warps and gaps in protoplanetary disks (Terquem et al. 1999/00, Bryden et al. 1999, ApJ, 514, 344)
  • Gravitational scattering as the source of interstellar clouds (Kegel et al. 1983 A&A, 119, 101; 1986, 161, 23; 1990, 232, 447, 461; 1990, 240, 123.)
  • Dynamical friction and removal of gas from young clusters (Saiyadpour, A., Deiss, & Kegel 1997, A&A, 322, 756)
  • Heating of the stellar galactic disk (Binney, J. M. 2000, see Jenkins, A. 1992, MNRAS, 257, 620 for history of problem and earlier references.)
  • N-body techniques-reviews in Highlights of Astronomy 1998, vol.11A, pp. 583 (Spurzen et al.) and 591 (Heggie et al.). Also Einsel, C. and Spurzen, R. 1999, MN, 302, 81).
  • Future tests of stellar dynamics from upcoming space missions (e.g. GAIA)
  • Gravitational waves from coalescing binaries, cosmic strings, etc.
  • The future of gravitational wave astronomy
  • Gravitational lens signatures of cosmological dark matter and other applicaitons


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17 January 2001
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