Tag Archives: kepler’s third law

The Teacher, the Law, and the Freshman

by Shane L. Larson

My mother has a theory — whatever your great passion is in the first grade, that is in all likelihood the best indicator of what you should do with your life. It’s what will make you the happiest.   For most of my years up through about fourth grade I wanted to be a scientist.  That changed to astronaut after the launch of the space shuttle Columbia in 1981.

The astronaut thing stuck for a long time, until I started college.  Growing up during the space shuttle era, I knew that to be an astronaut did not mean you had to be a pilot, because you could be a mission specialist — someone who did science on orbit.  At some point in the years leading up to college I had decided the thing that would maximize my chances of flying as a mission specialist was to be an engineer.  So the fall I went to college, my education did not start in science at all — it started in mechanical engineering.

Being an engineer was okay; we learned computer programming (in FORTRAN, unfortunately), we got to build balsa wood bridges and then crush them under a two ton load tester, and we got to peek under the hood of all kinds of devices and see how the world works when we’re not looking. I have a strong independent streak, and generally regard my destiny as my own to control. This caused a great deal of strife as a  young engineering student because I didn’t feel compelled to follow the list of courses that had been outlined for me. Unbeknownst to me, a battle was brewing over this fact.  I became suddenly aware of this near the end of my first winter quarter when the dean of engineering called me into his office.  My memory of the conversation is something like this:

  • DEAN: You are off track. You’re not taking the courses you are supposed to.
  • ME: So?
  • DEAN: You’re taking a random physics course. You can’t do that.
  • ME: Why? I want to take astronomy; I think it will be useful.
  • DEAN: The recommended courses are what is going to be useful.
  • ME: Well I’m going to take astronomy. I’ll get to those courses eventually.

In the end, I got to stay in the astronomy class, because spring quarter wasn’t too far away.  The dean did, however, make me sign a contract that said I would adhere to the recommended engineering schedule during all future quarters.

The late Dr. David Griffiths (Oregon State University, Department of Physics).

The astronomy course in question was Introductory Astronomy: Stars and Galaxies, taught by the late Dr. David Griffiths of Oregon State University (not to be confused with the other Dr. David Griffiths, at Reed College, of textbook fame).  Dr. Griffiths was one of the formative figures in my scientific youth, and is responsible for setting me on the path I am on today.  Three days after starting his astronomy course, I changed my major to physics (*).  While I gained some deep satisfaction from delivering my change of major paperwork to the College of Engineering, and even more satisfaction from tearing up the silly contract I had signed the quarter before, you must be wondering what it was that sparked such a drastic alteration in my destiny?  It was Dr. Griffiths.  And Johannes Kepler.

On the first day of astronomy class, I sat squirming in my seat, breathless with anticipation of learning about quasars, dark matter, and black holes. Dr. Griffiths strode into class, his salt-and-pepper curly black hair slightly wild and unkempt in a way that is typical for many scientists.  On that first day, he launched into a story about Johannes Kepler and the laws of planetary motion, which apply to all orbits not just to planets. This was a story I was familiar with from many long hours spent watching Cosmos (Ep. 3, “The Harmony of the Worlds”).  This wasn’t what I was exactly waiting for, but it was astronomy and so I enjoyed myself.

The story focused on Kepler’s Third Law, which tells us that the length of time a planet takes to complete an orbit is related to how big that orbit is.  This is called Kepler’s “Harmonic Law” and is usually stated as: “The square of a planet’s period (the time it takes to complete one orbit) is proportional to the cube (third power) of its semi-major axis (the radius of the orbit if it is circular).”  Mathematically, it is written as

P2 ~ a3

Kepler had deduced this result by studying careful observations of the positions of the planets made by his contemporary, Tycho Brahe.  This made perfect sense to me.  It was the way I had always been taught that science worked: you make observations, then deduce the Laws of Nature from those observations.

At this point in the story, Dr. Griffiths did something that changed my destiny forever.  He turned to us, and looking through glasses that had slipped to the end of his nose, said, “We could have figured that out even if we didn’t have any observations.”

I sat up in my chair a little straighter at this point. What?

Dr. Griffiths continued: “We can mathematically fit any data we want to — that’s what Kepler did. But whatever the fit is, it had better come from the fundamental Laws of Nature. We can derive Kepler’s Third Law from the basic rules of physics.”  Which he did.  With a deft hand, in white chalk, he completely blew my mind in just a few short lines.  That derivation is replicated below (in my own handwriting — this moment in my formative history is far too important to ever be reduced to mere typeset equations).  While it would be easy to include this just for the aficionados (and any engineers who think they may want to be physicists…), you should look closely at this.  This is the beauty of the Laws of Nature.  What makes the planets go, and why they move the way they do, was once one of the greatest mysteries of science, yet through sheer force of imagination we humans figured out how to explain it concisely enough that it can be written on a napkin and explained as a motivational exercise for students!

Derivation of Kepler’s Third Law from first principles.

This was a revelation to me.  It was the first time that I truly felt I had a glimpse into what the scientific enterprise was all about.  It was the first time that I had ever encountered the awesome power of science, and the first time I had ever been confronted by the sheer scope of what we could accomplish with our naked intellect.  But most importantly, in the span of those few minutes, as the scritch scritch of Dr. Griffiths chalk outlined the foundations of Kepler’s Third Law, I learned one of the most valuable lessons of my scientific career, which still guides me today: everything is connected, and knowing as much as possible about everything you possibly can will bear unexpected and beautiful fruit in the end.  Before that class was over, I had decided that if this is what physics was all about, then this was a profession for me.  I walked out that day and started figuring out how to change my major.

The Laws of Nature would exist whether you and I were sitting here talking about them or not.  Atoms would continue to bond together and make stuff, apples would continue to fall out of trees and to the ground, and stars would continue to burn and flood the Cosmos with light.  The fact that the Laws of Nature can be figured out is one of the great gifts of the Cosmos to its inhabitants.  We live in a world filled with predictable patterns.  Rocks that I throw up in the air always fall back to the ground. A cup of coffee on my counter always cools to the ambient temperature of the room.  The constellations rise in the east each night at dark, and slowly march across the sky westward over the course of the night.  These tantalizing patterns are clues that Nature provides us about the underlying order of things, breadcrumbs that lead us to science, the express goal of which is to understand the Laws of Nature that make all the wondrous order of the natural world.

Science is a unique creation of our species, the paramount expression of our ability to see the world around us, and imagine why it is the way it is.  The culmination of our creative ruminations are the Laws of Nature, written in the language of mathematics, another invention of our species. As we practice science today, it is a self-correcting framework.  At any given moment, it captures our imperfect understanding of what we have observed about the Cosmos. When we make new discoveries, we reexamine the Laws of Nature as we understand them, and expand our thinking to correctly explain what we have seen.   Johannes Kepler published his Harmonic Law in 1619.  It was perfectly adequate for explaining observations of the planets in the solar system.  But it was not until 1687 that Newton took that understanding and expanded it using the Universal Law of Gravitation.  I’m sure he didn’t imagine that the consequence of that simple expansion of human consciousness would be to create a scientist out of a university freshman 300 years later.

Dr. Griffiths passed away in early 2005, returned to the embrace of the Cosmos from whence we all came.  It is a great sadness to me that I never made it back to Oregon State to talk to him again after I graduated.  Teaching is an artform which requires immense amounts of patience and practice.  I think back on that day in Dr. Griffiths’ class often, deeply cognizant of the fact that that was the moment the changed my career forever.  Today, I teach my own classes, and I hope that I also occasionally stumble through moments of revelation for my students.  I don’t know what those moments might be, or when they might occur.  I just hope they happen.  But I take my cues from Dr. Griffiths: I always teach the derivation of Kepler’s Third Law, just like it was taught to me.  🙂


(*) This is how I always tell this story, because that is the way I remember it in my head. However, having written it down and looking at it critically, it does seem the exact time that passed between events is not relayed accurately here. However, this is my story, and I’m sticking to it. 😉