# ASTR 241: Class Log

### 1. Solar System Overview

The first week introduces key elements of the Solar System, including the Sun, terrestrial and giant planets, smaller bodies, and the solar environment. Some useful background material is covered in Chapters 1.3 (Celestial Motions), 2.3 (Copernican Astronomy), 2.4 (Galileo: The First Modern Scientist), and 8.1 (Two Types of Planets). Kepler's laws are introduced as a comprehensive (albeit approximate) description of planetary motion. They are covered in Chapter 2.5 (Kepler's Laws of Planetary Motion); some important properties of the ellipse are covered in Chapter 3.1 (Deriving Kepler's Laws; see last paragraph on p. 68 through equation (3.42) on top of p. 72).

Handout: Syllabus, Discussion Question.

Board: Orbital Motion Before Newton, Geometry of Ellipse (Drawing an Ellipse [src]), Kepler's Laws (Law III [Fig. 2.19]), Falling into Orbit [24 Aug], Questions #1, #2, Questions #3, #4 [26 Aug].

Slides: Solar System Overview [22 Aug].

Assignment: Problem Set #1: Due 29 Aug 2016; Answers.

### 2. Kepler's Laws Derived

This week covers the derivation of Kepler's Laws from Newtonian mechanics, a brief discussion of conic sections, and Kepler's Equation. In the book, Kepler II (equal areas in equal times) is derived in Chapter 3.1.1, Kepler I (orbits are ellipses with Sun at one focus) is derived in Chapter 3.1.2, and Kepler III (P2/a3 = constant) in Chapter 3.1.3. We followed the same order, but not always the same strategy. We derived Law II using Cartesian coordinates, and then transformed to polar coordinates. We derived Law I by introducing a new variable u = 1/r and used our results for Law II to eliminate the time t in favor of the angle θ; the resulting differential equation resembles the harmonic oscillator, and the solution, restated as a function r(θ), is identical to the book's equation (3.34). Our approach to Law III, on the other hand, closely followed the book's.

Kepler's equation is not covered in the book, but it is necessary to calculate the actual position of a planet or other orbiting object at a given time t. Our derivation relied on elementrary geometry; Fig. 2 in Problem Set #2 is a good resource for this derivation. Kepler's equation is non-linear, and a numerical method is required to solve it; we briefly covered Newton's method for finding solutions to the equation f(x) = 0.

Assignment: Problem Set #2: Due 7 Sep 2016; Answers.

### 3. Orbital Motion

This week wrapped up our discussion of one-body and two-body orbital mechanics. We began with the external and internal gravitational fields of spherical bodies, and used Gauss's Theorem to show that the external field of a spherical body of mass M is equal to the field of a point of the same mass M located at the same position. We also showed that the internal field of a spherical shell of matter is zero. (These points are not covered in the text.) Building on the results of Problem Set #2, Question #1, we introduced simple generalizations of Kepler's Laws for two-body systems (in which both bodies have comparable masses). We then covered the concept of orbital energy, and showed that it is conserved; the book covers this in Chapter 3.2. Finally, we briefly discussed the Virial Theorem, which the book covers, in much more depth, in Chapter 3.4.

Assignment: Problem Set #3 (just do Questions #1 and #2): Due 12 Sep 2016; Answers.

Links: Centrifugal Force [xkcd's take on pseudo-forces].

### 4. Thermal Equilibrium

This week covered basic ideas of thermal equilibrium applied to planets, as discussed in Chapter 8.2. We began with thermal radiation, sketching black-body curves (equation 5.90) and stating the Wien (equation 8.4) and Stefan-Boltzmann (equation 5.96) laws. We used the latter to calculate the Sun's luminosity (equation 8.3), and energy conservation to derive the solar flux (equation 8.5). Equating solar heating with radiative cooling, we derived expressions for the equilibrium surface temperature of a planet (equations 8.9 and 8.10). A similar approach, invoking thermal equilibrium for both the surface and the atmosphere, explains the greenhouse effect.

The motion of gas atoms provides another example of thermal equilibrium. We start with the fact that the average energy of motion in 1-D is (1/2)kBT,where kB is Boltzmann's Constant. Collisions establish the Maxwell-Boltzmann distribution (equation 5.40) of molecular velocities with root-mean-square (rms) speed vrms = √3kBT/m, where m is the mass of a gas molecule (equation 5.48). Molecules can escape a planet if their speeds exceed the planet's escape velocity, vesc = √2GM/R; molecular collisions repopulate the high-velocity "tail" of the Maxwell-Boltzmann distribution, so a simple criterion for atmosphere to escape is vesc < 6 vrms (page 202). In practice, however, a fast-moving molecule will only escape if its mean free path is long enough; this condition is satisfied on the surface of the Moon, but not on the surface of the Earth.

Assignment: Problem Set #4: Due 21 Sep 2016 [note revised due date]; Answers.

Links: Energy flow in atmosphere (Halpern et al. 2010, Int. J. Mod. Phys. B, vol 24, 1309—1332). A Timeline of Earth's Average Temperature [xkcd's summary of Earth's temperature].

### 5. Hydrostatic Equilibrium

This week covered hydrostatic equilibrium, convection, and the structure of planetary atmospheres. The equation of Hydrostatic Equilibrium is covered in Chapter 9.2. However, the book only discusses convection in the context of stellar structure; to find a treatment similar to the presentation in class, see Chapter 15.1.3. Finally, the temperature profile of the Earth's atmosphere is shown in Figure 9.5, and circulation patterns for nonrotating and rotating planets are shown in Figure 9.6.

Assignment: Problem Set #5: Due 28 Sep 2016 [note revised due date]; Answers.

### 6. Review #1

Board: Problem Set 4, Questions #1, #2, Review Questions #5, #6, Problem Solving Tips [26 Sep] Problem Set 5, Question #3, Review Question #7, Question #2, Question #4 [28 Sep], Midterm Discussion [3 Oct].

Handout: Review Questions #1.

### 7. Terrestrial Planets

This week focused on the internal structure of terrestrial planets, on their internal heat sources, and on the consequences of heat flow.

Slides: Terrestrial Planets [5 Oct].

Assignment: Problem Set #6: Due 12 Oct 2016; Answers.

### 8. Giant Planets

Slides: Giant Planets [11 Oct].

Assignment: Problem Set #7: Due 19 Oct 2016; Answers.

Links: Jupiter Submarine [another phase diagram], Jupiter Descending [clouds and more clouds].

### 9. Earth, Moon, and Sun

Board: Tidal "Forces", Ocean Tides. I, Ocean Tides. II [17 Oct], Tidal Torques, Spin-Orbit Evolution, Precession, Precession of Moon's Orbit [19 Oct].

Slides: Earth, Moon & Sun.

Assignment: Problem Set #8: Due 26 Oct 2016; see images of Moon below. Review.

### 10. Satellites & Rings

Board: PS#7, Question #3, Roche Radius [21 Oct], Nearly–Circular Orbits, Mean–Motion Resonance, Tidal Heating in Io [26 Oct], Non–Spherical Satellites, Question #1, Question #2 [28 Oct].

Slides: Satellites of the Giants [21 Oct], Rings [26 Oct].

Assignment: Problem Set #9: Due 2 Nov 2016 (note corrections to Question #2); Answers.

### 11. Asteroids, TNOs, & Comets

Board: Asteroids & TNOs, Orbits, Lagrange Points [31 Oct], Comets: "Dirty Snowballs", Tails, Outgassing Rate [2 Nov], Question #1, Question #2(b), Question #2(d) [4 Nov].

Slides: Asteroids & TNOs [31 Oct], Comets [2 Nov].

Assignment: Problem Set #10: Due 9 Nov 2016; Answers.

### 12. Topics: NEOs & ESPs

Board: Near-Earth Objects, Impact Physics, Detection & Deflection [7 Nov] Formation of Solar System (note marks in checkboxes added later), Detection of Extra-Solar Planetary Systems, Results [9 Nov].

Slides: Near-Earth Objects [7 Nov], Other Planetary Systems [9 Nov].

### 13. Review #2

Handout: Review Questions #2.

### 15. The Sun. II

Board: Solar Models: Equations, Radiative Zone [28 Nov], Solar Atmosphere [30 Nov], Wave-Particle Duality, Bohm Model of Hydrogen, Spectral Lines [2 Dec].

Slides: Solar Models [28 Nov], Astronomy Scholarships [2 Dec].

Links: The Solar Spectrum, Spectrum Image [30 Nov].

### 16. Review #3

Board: Question #6, Question #13, Question #10 [5 Dec], Question #23, Questions #17, #14, Physics Themes [7 Dec].

Handout: Review Questions #3.

Extra Credit: Anybody may submit answers to the last four review questions before 5 pm on Friday, 9 Dec and receive feedback on the answers before the final exam. Students who got scores below the average on both midterms (midterm 1: 65 out of 90 points; midterm 2: 60 out of 90 points) will get up to 15 points of homework credit per problem. Answers may be submitted by email.

 Joshua E. Barnes      (barnes at hawaii.edu) Updated: 8 December 2016 http://www.ifa.hawaii.edu/~barnes/ast241_f16/classlog.html