Astronomy 250: Special Topics in Astrophysics

Fall 2011

Planetary Dynamics

Mechanical orrery: a clockwork model of the planets orbiting the Sun

Gravitational dynamics of planetary bodies, both in the solar system and in extrasolar systems. To develop intuition, we will begin with classical analytic descriptions of the N-body problem. We will then introduce modern numerical integration techniques. Having developed an understanding of point-mass interactions, we will impart shapes to planetary bodies to study tides and spin-orbit coupling. The goal is to understand the literature quantitatively.

Course information

Instructor:
Eugene Chiang (Departments of Astronomy and of Earth and Planetary Science)
Time & Place:
Fridays, 1:00–4:00 p.m., 401 McCone.
Format:
Weekly lectures, occasional problem sets, and eventually round-table discussions of the literature.
Requirements:
(a) Make a reasonable attempt on at least one problem per problem set. (b) When discussing papers in class, be in a position to derive (at the order-of-magnitude level) equations and explain their physical significance. (c) Final project to write a numerical integrator.
Office Hours:
Thursdays, 2:30–3:30 p.m. (Hearst Field Annex D29B or adjacent conference room D29D), and also anytime I am around and not talking with anybody else. For guaranteed meeting times, email me (echiang [at] astro.berkeley.edu).
Text:
Solar System Dynamics by Murray & Dermott (MD), plus selections from Binney & Tremaine’s (BT) Galactic Dynamics, plus journal articles.

Course Outline

I. Tour of the Mechanical Universe

II. The 2-Body Problem

  1. Problem Set 1 (due Friday Sep 9; also reading assignments from MD and BT) (PDF) — Solutions to PS 1 (PDF)
  2. Chapter 3 of BT (only certain sections are relevant; see Problem Sets 1 and 2) (PDF)
  3. Osculating elements — Notes on osculating elements and Gauss’s perturbation equations (PDF)
  4. Useful relations
  5. Guiding center description
  6. Kicking the 2-body problem: Gauss’s perturbation equations — Lecture 1 slides on kicking orbits (PDF)

III. The Restricted 3-Body Problem

  1. Problem Set 2 (also due Friday Sep 9; also reading assignments from MD and BT) (PDF) — Solutions to PS 2 (PDF)
  2. Chapter 8 of BT (only certain sections are relevant; see Problem Set 2) (PDF)
  3. Hill’s equations and the Hill sphere (Roche lobe)
  4. The Jacobi constant
  5. Lagrange points
  6. Horseshoes and tadpoles
  7. Reading 1 (due Sep 16): “Towards a theory for the uranian rings” by Goldreich & Tremaine 1979.
  8. Reading 2 (also due Sep 16): “On the width and shape of gaps in protoplanetary disks” by Crida et al. 2006 (read just enough to understand Figures 1, 2, 3, and 5).
  9. Supplemental Reading 1 (also due Sep 16): “A method for calculations of nonlinear shear flow: Application to formation of giant planets in the solar nebula” by Korycansky & Papaloizou 1996 (relevant for question 9).
  10. Supplemental Reading 2 (also due Sep 16): “Gas flow across gaps in protoplanetary disks” by Lubow and D’Angelo 2006.
  11. Supplemental Reading 3 (also due Sep 16): “Dynamics of narrow rings” (PDF) by Dermott 1984 (might be relevant for question 8).
  12. Sign-up sheet for homework related to readings (due Sep 16)
  13. Sep 23 — Reading 3 (Part I): “Planet Formation by Coagulation: A Focus on Uranus and Neptune” by Goldreich, Lithwick, & Sari 2004 (Part I = Sections 1–6, everything but skim discussion surrounding equations 34–35).
  14. Sep 23 — Sign-up sheet for homework related to Part I
  15. Sep 30 — Reading 4 (Part II): “Planet Formation by Coagulation: A Focus on Uranus and Neptune” by Goldreich, Lithwick, & Sari 2004 (Part II = Sections 7–12).
  16. Sep 30 — Sign-up sheet for homework related to Part II
  17. Sep 30 — Supplemental Reading: “The Formation of Ice Giants in a Packed Oligarchy: Instability and Aftermath” by Ford & Chiang 2007 (relevant to the onset of instability in oligarchies).
  18. Sep 30 — Supplemental Reading: “The Largest Kuiper Belt Objects” (PDF) by Brown 2008 (“about 10 Plutos in the Kuiper belt”).

IV. The (Very) Disturbing Function

  1. Strengths and arguments
  2. Secular, resonant, and short-period terms
  3. Lagrange’s equations
  4. Problem Set 3 (due October 14) (PDF) — Solutions to PS 3 (PDF)

V. Mean-Motion Resonances

  1. Good and bad phasing
  2. Pendulum model
  3. Libration periods and maximum widths
  4. Reading 5 (due October 21): “On Detecting Terrestrial Planets with Timing of Giant Planet Transits” by Agol, Steffen, Sari, & Clarkson 2005.
  5. Oct 21 — Sign-up sheet for homework related to Reading 5
  6. Resonant capture and migration
  7. Reading 6 (due October 28): “The California Planet Survey. III. A Possible 2:1 Resonance in the Exoplanetary Triple System HD 37124” by Wright et al. 2011.
  8. Reading 7 (also due October 28): “Architecture and Dynamics of Kepler’s Candidate Multiple Transiting Planet Systems” (just Section 5) by Lissauer et al. 2011.
  9. Reading 8 (also due October 28): “Origin of the Orbital Architecture of the Giant Planets of the Solar System” by Tsiganis et al. 2011.
  10. Reading 9 (also due October 28): “Resonant Capture by Inward-Migrating Planets” (Figures 1, 2, 4, and 5 only) by Yu and Tremaine 2001.
  11. Also due October 28 — Sign-up sheet for homework related to Readings 6–9
  12. Problem Set 4 (also due October 28) (PDF) — Solutions to PS 4 (PDF)

VI. Secular Theory

  1. Wires
  2. Laplace–Lagrange theory for eccentricities and inclinations
  3. Problem Set 5 (due November 18) (PDF) — Solutions to PS 5 (PDF)
  4. Kozai oscillations
  5. Reading 10 (due December 2): “Secular Chaos and the Production of Hot Jupiters” by Wu & Lithwick 2011.
  6. Reading 11 (due December 2): “Theory of Secular Chaos and Mercury’s Orbit” by Lithwick & Wu 2011.
  7. Optional Reading 12 (due December 2): “Chaotic Diffusion in the Solar System” by Laskar 2008.
  8. Also due December 2 — Sign-up sheet for homework related to Readings 10–12