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SPRING 2001
PROFESSOR:
John Scalo
Office: R.L. Moore 17.220
Phone: 471-6446 (office), or 478-2748 (home)
Email: parrot@astro.as.utexas.edu
Description and Tentative Syllabus
1. SOME HISTORY
"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.
2. LEVEL OF PRESENTATION
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.)
3. ASSIGNMENTS AND GRADING
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.
4. TEXTBOOKS
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.
5. THE SYLLABUS
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)
Chaos
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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