Order of Magnitude Astronomy Astr 734 — Fall 2017 Wednesdays 2:30

Order of Magnitude Astronomy Astr 734 — Fall 2017 Wednesdays 2:30 — 3:30pm John Tonry

1 Introduction

Estimation is an essential skill for astronomy and life in general. This course will introduce students to the utility of order of magnitude calculations and practicing the ability to “think on your feet”. I will give short overviews of basic physical concepts but the majority of the class time will be spent interactively, with students tackling problems at the whiteboard. Problem sets will be pre-assigned with questions similar (and sometimes identical) to some in the Ohio State course. If time permits, I will challenge the class by extending the problem or asking a new, unseen question. Learning OOM is like learning to swim - you cannot succeed by watching from dry land. Grading is therefore largely based on participation. Showing up and being ready to answer the pre-assigned questions will account for 80% of the grade. If you cannot attend class because of a good reason (NOT because of a pressing deadline for something else), please let me know in advance. Students will submit their own order-of-magnitude astrophysics question and answer for the remaining 20% of the grade. We will use the Astr 734 wiki heavily throughout this course. Student’s contributions to the wiki constitute part of their participation. Related courses and reading:

• Cheatsheet of useful numbers (constants, conversion factors) from Tie- len’s ISM book.

• Ohio State course (OSU)

• Astrophysics in a Nutshell by Dan Maoz for a good introduction to the various astrophysical concepts we will be using

• Eugene Chiang’s Order of Magnitude Physics course (Berkeley)

1 • Sterl Phinney’s Order of Magnitude Physics course (Caltech) • Dave Jewitt’s Order of Magnitude Earth and Planetary Sciences course (UCLA) • Order-of-Magnitude Physics: Understanding the World with Dimen- sional Analysis, Educated Guesswork, and White Lies (Goldreich, Ma- hajan, & Phinney) • Order of Magnitude Physics: A Textbook with Applications to the Retinal Rod and to the Density of Prime Numbers (Mahajan PhD thesis) • The Astronomical Reach of Fundamental Physics (Burrows & Ostriker 2014) Our web page can be reached at http://www.ifa.hawaii.edu/users/jt/astr734 and the wiki is found at http://astr01.ifa.hawaii.edu/ay734.

2 Why do OOM?

The whole point of being a scientist is to discover new things. The only way that happens is to think up new ideas and see if they pan out. It is certainly true that 90% (or 99% or 99.99%) of ideas that anybody comes up with do not work. They may be obviously ﬂawed, subtly ﬂawed, already known, not useful or important or interesting in actuality, or they may be pure gold. How in the world do you decide? That’s what OOM is really all about. It’s useful • Because it allows you to quickly evaluate new ideas and discard blind alleys. • Because it gives you insight and understanding into “what’s really going on” before sweating the details. • Because it’s very broadly applicable: it’s part of being “numerate” and being able to make good decisions throughout your life. • Because it’s useful for your quals. The faster and better you can do OOM, the faster you can sift through the 99% of bad ideas and work on something good.

2 3 How the course will work

I will assign two questions each week and collect solutions at the start of each class. A student will be selected at random to answer the question, or possibly a brand new question! (Possibly) I will ask a student to go to the board and give an OOM explanation of the most recent colloquium. Each student will contribute four OOM questions and their best stab at a solution throughout the semester, due roughly once a month. Grading will be 80% based on participation in the class, contribution to the wiki, and quality of solutions. The remaining 20% will be based on the quality and solution of the student questions. The grading is absolute: ba- sically thumbs up, neutral, or thumbs down in the opinion of the instructor (and possibly the class) based on whether the answer was clean and work- manlike, vague or rambling, or just awful and wrong. (Of course we all suﬀer from the latter from time to time — we’re just trying to learn how to avoid it as much as possible!) You should work independently on the homework. You may consult with the instructor or other faculty or postdocs, but not your fellow classmates. Obviously this should not involve a computer, and to the greatest extent possible try not to look anything up until you’ve gotten just as far as you can. OOM is like swimming — you want to get rid of the water wings as soon as you can. You should show up to class every week, rain or shine. Absence that does not aﬀect your grade is permitted only for very good reasons provided ahead of time: observing run, conference, etc. You are still responsible for delivering your written solutions prior to class.

4 The wiki

We will use the Astr 734 wiki extensively as a place to write down constants, conversion factors, formulae, data tables, etc. We may post questions and answers there, we may post pictures of the whiteboard.

3 5 Resources

Similar courses are taught at Berkeley, OSU, Caltech, etc. There is a book by Mahajan that is very instructive, you should read it. Various general grad-level astrophysics texts exist, such as “Modern As- trophysics” by Carroll and Ostlie, or “Astrophysics in a Nutshell” by Maoz.

6 OOM Strategies

The basic repertoire of order-of-magnitude techniques, quoted directly from Mahajan: • Divide and conquer: Split a complicated problem into manageable chunks, especially when you must deal with tiny or huge numbers, or when a formula naturally factors into parts (such as V ∼ l × w × h).

• Guess: Make a guess before solving a problem. The guess may suggest a method of attack. For example, if the guess results in a tiny or huge number, consider using divide and conquer. The guess may provide a rough estimate; then you can remember the ﬁnal estimate as a correc- tion to the guess. Furthermore, guessingand checking and modifying your guessimproves your intuition and guesses for future problems.

• Talk to your gut: When you make a guess, ask your gut how it feels. Is it too high? Too low? If the guess is both, then its probably reliable.

• Lie skillfully: Simplify a complicated situation by assuming what you need to know to solve it. For example, when you do not know what shape an object has, assume that it is a sphere or a cube.

• Cross-check: Solve a problem in more than one way, to check whether your answers correspond.

• Use guerrilla warfare: Dredge up related facts to help you make an estimate.

• Punt: If youre worried about a physical eﬀect, do not worry about it in your ﬁrst attempt at a solution. The productive strategy is to start estimating, to explore the problem, and then to handle the exceptions once you understand the domain.

4 • Be an optimist: This method is related to punt. If an assumption allows a solution, make it, and worry about the damage afterward.

• Lower your standards: If you cannot solve the entire problem as asked, solve those parts of it that you can, because the subproblem might still be interesting. Solving the subproblem also clariﬁes what you need to know to solve the original problem.

• Use symbols: Even if you do not know a certain value — for example, the energy density stored in muscle — deﬁne a symbol for it. It may cancel later. If it does not, and the problem is still too complex, then lower your standards. Some other things that you really should appreciate. • You’ve got to memorize some stuﬀ. Physical constants and conversion factors for sure. Basic physics equations for sure. There is no way around this. No pain, no gain. Numbers matter.

• Intermediate between an equation in terms of symbols and a numerical calculation lies an extremely useful midpoint: the scaled equation. For example:

3 2 ρ R v E = 100Mton 2g/cm3 100m 10 km/s is the kinetic energy of a sphere of density ρ and radius R traveling at velocity v. By scaling variables to useful units for asteroids and using an output unit that may be directly meaningful in terms of impact damage, we prime our intuition and we make an equation that can be readily combined with others for an OOM estimate.

• The two things you want to avoid like the plague are a) simply freezing and giving up — you can always say something useful if you “lower your standards”, and b) maundering on and on about tactical details without stopping and plotting a strategy.

7 Example problem

How long will it take for humans to colonize the galaxy?

5 8 For next week

• Log in to the wiki and start populating it with things you think are useful. We’ll want to core dump but also organize things, so do this thoughtfully as you learn how trac wiki works.

• Read or at least skim the ﬁrst six chapters of Mahajan.

• People talk about using a “gravity tractor” to deﬂect an asteroid. This is simply a spaceship that very gently thrusts while staying gravita- tionally bound to an asteroid. Obviously you can’t use much thrust, but it does avoid the problem of trying to use a rocket attached to a wildly spinning asteroid. Why not use an “electric tractor”? Suppose the spacecraft has a par- ticle accelerator that ﬁres a beam of charged particles into the asteroid until the electrostatic force between the spacecraft and the asteroid be- comes large. Could this work as well or better than using the force of gravity?

• Try thinking of whether a “magnetic tractor” could work, based on modern, strong permanent magnets.