From the Earth to the Moon

I recently finished reading The Birth of a New Physics, by I. Bernard Cohen, which describes the 17th century transition from Aristotelian to Newtonian physics. This reminded me of a demonstration I did for my astronomy sections last semester, in which I tried to impress them with the power of Newtonian unification. (It didn't work.) And yesterday was the day we celebrate projectile motion, so that's as good an excuse as any to revisit the topic.

As I mentioned in my last post, I think we suffer from presentism that makes it difficult for us to understand how our predecessors saw the world. To remedy that, I've been reading a lot of history of science recently; I want to understand the role that science has played in changing our conception of the world.

When reading history of science, I sometimes struggle with the seemingly glacial pace of scientific advances that I, with my present level of education, can work out in a few lines. I am no genius, so why did it take humanity's greatest scientific minds generations to find the same solutions? The answer is these solutions originally required deep conceptual shifts that for me—thanks to the work of those scientists—are now completely in the background. Here's an example that I think simultaneously demonstrates the power of Newtonian analysis and the elusiveness of the modern scientific perspective.

Aristotelian physics held that everything from the moon up moved only in circles and was perfect and unchanging, while everything below the moon was imperfect, impermanent, and either drawn toward or away from the center of the universe. The critical thing is that the motion of objects on earth—projectiles, boats, apples—operated according to fundamentally different rules than the motion of stars, planets, and other celestial objects.

What Newton did was to show the same rules apply everywhere, to everything. His laws of motion and gravity work for cannon balls, birds, the moon, and even once in a lifetime comets. This is where our presentism hurts us, because that radical idea seems completely obvious now. Of course physics underlies both airplanes and space probes. Duh.

In the abstract, that's an easy case to make. But the demonstration I did in class, which is a modern-ish take on an analysis Newton himself performed, might be able to show how cool and counterintuitive this unification really is.

Consider this: if you drop a rock from a given height and time its descent, you can explain why a month is roughly 30 days long. These two facts seem completely unrelated but turn out to be connected by a simple law.

Aristotelian physics says that heavy objects are naturally drawn toward the center of the universe and that the celestial moon naturally moves about the Earth in a perfect circle. But even ignoring the Aristotelian perspective, from our modern vantage the link between these two facts seems kind of incredible. We have some vague idea that the length of a month is connected to the cycles of the moon, and we know that gravity makes rocks fall, but the moon is clearly not falling and rocks have nothing to do with calendars; so how are these facts related?

Now, I'm not shocking anybody by saying that gravity is the common factor, but I want to show you how relatively simple it is to work this out using the tools Newton gave us.

Newton's law of universal gravitation says that gravity is an inverse square force. In fact, other scientists before Newton (Kepler, Hooke) had suggested this. It was known that the intensity of light falls off with the square of distance; maybe the same principle worked for gravity, too. Force is proportional to acceleration, so you can measure it by timing falling objects (or the period of a pendulum, which was the most precise method available during Newton's time). At the surface of the earth, this is 9.8 m/s² and usually denoted with a g.

If the earth is also pulling on the moon, and gravity is an inverse square law, we can find out how much earth's gravity is accelerating the moon. Divide the distance to the moon by the radius of the earth (figures known since the ancient Greeks), square the result, and that's how much weaker gravity's action on the moon is.

The distance to the moon is about 60 times the radius of the earth, so earth’s gravity pulls on the moon with 1/3600 the force that it pulls on a rock near the surface. But even so, shouldn't the moon be here by now? It's obvious that the moon is circling the earth and not slamming into us.

What we need here is another law. We see circular motion on earth, too. Imagine tying a string to a rock and spinning the rock around. What keeps the rock moving in a circle? The string, which is taut. The string pulls on the rock so that it doesn't go flying off. But if the string is pulling the rock inward, why doesn't the rock come inward toward your finger? Well, imagine slowing down the spin rate of the rock. Do that and the whole thing will fall limp. There is a specific speed required to keep the string taut. In fact, if you spin too fast, the string will break and the rock will fly off.

So here's the law. When considering circular motion, inward (centripetal) acceleration is equal to the square of the spin rate (angular velocity) times the radius. The faster you spin the rock, the harder the string needs to pull on it to keep it from flying off.

If we assume the moon is going around the earth in a perfect circle, and we suppose that gravity is pulling it inward at 1/3600 the strength it does on earth's surface, then we can figure out the moon's spin rate (around the earth), too. A little algebra gets us this formula:



r_e is the radius of the earth. The angular velocity ω is how many radians per second the moon moves. To figure out how many seconds it takes to make a single orbit, you basically just flip the expression upside down and multiply by 2π to get a full circle. That gives you:



Plug in the right numbers (r_e=6378 km, g=9.8 m/s²) and you arrive at a t of about 2.35 million seconds, which comes out to roughly 27.3 days (the sidereal period).

This is a couple days off from 29.5 days, which is how long it takes the moon to go through a complete set of phases (the synodic period). The difference is due to the fact that after those 27.3 days, the earth has also moved about 1/13 of the way around the sun, changing where the sun is in the sky. Because the phase of the moon arises from its position relative to the sun, it takes the moon a couple more days to catch up with the sun’s new position.

Those complications aside, the ease with which you can find the moon's sidereal period from a measurement of surface gravity is both stunning and surprising. The calculation is literally only a few lines long. Here, look for yourself:

Credit: Me me me

I'm not showing you this to impress you with my mathematical talent, but to bring you back to my initial perplexity. Why did it require an intellectual titan such as Newton to figure this out? That is, what conceptual leaps were necessary? I don't know that I can answer that question completely, but here's a partial explanation that comes in large part from Cohen's book.

First of all, as I've said, Newton had the creativity and imagination to suggest a unified physics at all. Others at the time were formulating laws that applied to the heavens (Kepler's laws of planetary motion) and even physical mechanisms by which the planets moved (Descartes' vortices), but none imagined that a single law lay behind falling apples, the tides, planetary orbits, the moon's phases, the movement of Jupiter's satellites, and the orbits of comets.

Furthermore, Newton's laws of motion serve as a starting point for conceptualizing the moon's orbit. Aristotelian physics held that circular motion was perfect because celestial objects could return to their starting point indefinitely, continuing the motion for all eternity. Circular motion required no further explanation.

But Newton's first law says that objects have inertia, that they will continue in straight lines (or remain motionless) unless acted on by an outside force. This law isn't a formula but a tool for analysis. If you assume it is true, then you can look at any physics problem and immediately identify where the forces are. Thus, we can look at the moon, see that it is not moving in a straight line, and conclude there must be some force acting on it.

As I mentioned before, others had already proposed an inverse square law to explain gravity. Simply writing down the law of universal gravitation was not Newton's accomplishment. Instead, what Newton did was to prove mathematically that a body obeying Kepler's laws of planetary motion must be acted on by an inverse square force and the converse that an inverse square force will always produce orbits that resemble the conic sections (circles, ellipses, parabolas, or hyperbolas).

The proof Newton develops is heavily geometrical and begins by looking at an object moving freely through space that is periodically pushed toward a central focus. Newton then reduces the time between impulses until the force becomes continuous and the orbit, which began as a gangly polygon, curves into an ellipse. The important aspect here is there are two components to an orbiting body's motion: a central force acceleration and a velocity tangent to that acceleration.

What this means is the moon is falling toward the earth just as surely as an apple is. The difference is the moon is also moving in another direction so quickly that it continually misses the earth. This is what it means to orbit. As Douglas Adams said, "There is an art to flying, or rather a knack. The knack lies in learning how to throw yourself at the ground and miss."

Credit: Newton Newton Newton

All this groundwork (and more) was necessary so that Newton could justify a key step in those few lines of math I showed you up above. (I should point out that Newton's work didn't look anything like mine, because the notation and norms of math were very different back then.) The key step is that I equate the moon's acceleration due to gravity (a_m) with the centripetal acceleration of uniform circular motion (a_c). While the units are the same, a priori there's no reason to think the two are related.

Without a mathematical and physical framework detailing how mass, force, and gravity interact, equating those two conceptions of acceleration is nothing more than taking a wild guess. And if you're guessing, that means there are probably plenty of other guesses you could have made as well. This is what our presentism—replete with all the right guesses—hides from us. At each moment when a scientist does what comes naturally to us now, they had innumerable other options before them. The achingly slow pace of scientific discovery, then, is a result of all the frameworks and ideas and theories leading to those other guesses, equally valid a priori, that turned out not to be right.

As I've written before, in physics it is sometimes easy to guess the right answer. What I hope this post does is demonstrate that guessing—that moment of eureka when the correct answer finally materializes—is only the proverbial tip of the iceberg when it comes to science. This is important to remember when you think you’ve been struck by inspiration and arrived at a brilliant new truth about... whatever. Our popular conception of history valorizes those moments, but a fuller understanding of history vindicates the slow, haphazard, incremental work that must come first. If that work isn’t there, maybe your new truth isn’t, either.