Less Is More

The eclipse is only a couple days away, so I've been checking weather reports for the last week or so hoping the forecast will be clear. It varies from site to site and day to day, which is kind of frustrating.

Our inability to precisely predict the weather reminds me of some standard opposition to climate science. That is, if we can't even predict next week's weather in one city, how can we possibly predict the world climate a decade from now? There's a similar argument against evolution: if there's a single "missing link," how can we possibly claim to know the history of life? If we don't know the exact sequence leading from, say, our common ancestor with chimpanzees to anatomically modern humans, how can we be sure that life in general underwent evolution?

The problem with this line of thinking is that it misses why science has managed to be successful at all. Science does not make accurate predictions because we have perfect knowledge of a system. In fact, the opposite is often the case. Science succeeds in part because of our ability to abstract away that which is unimportant and reveal the underlying patterns. Consequently, a little bit of ignorance helps us miss those details which might distract us.

The eclipse itself is a great example of how having all the information can (literally and figuratively!) blind us. Our eyes are not well equipped for looking at the sun because it can be many orders of magnitude brighter than everything else we see. To fit a scene with such drastically varying brightness levels into our head, we lose some contrast resolution and end up not being able to see dim, feeble objects—especially those in the sky. That means we miss out on all the stars up during the day as well as anything even remotely near the sun.

During a total solar eclipse, the conveniently sized moon perfectly blocks the disk of the sun and a local night falls. The stars and planets come out, and the wispy corona that wreathes the sun materializes before our eyes. If we have good enough telescopes, we can measure the deflection of starlight around the sun, which tells us how its mass perturbs space itself.

1919 Solar Elcipse. This picture is in the public domain, but I guess I'll credit Arthur Eddington?

And yet we can only do this because we have less information, because we are no longer being blinded by the flood of photons.

Now perhaps I'm engaging in some rhetorical trickery here. I started off talking about how missing individual pieces of the puzzle doesn't prevent science from abstracting away the pieces to find the underlying rules, and then I shifted to discussing how having all the pieces hinders us. But I do think there's a connection here, because the truth is we don't always realize when the details have led us astray; it's not usually as obvious as the blazing sun.

The problem has to do with our affinity for patterns and the mathematical tools we've developed for describing them. As the history of geocentrism demonstrates, our tools are too powerful for their own good. Because planetary orbits are pretty complicated, geocentrists employed epicycles—circles on top of circles—to describe how the planets moved through the sky. (Copernicus did this as well, actually, because he couldn't give up the notion of perfectly circle orbits at constant speed.) Add enough epicycles to your system and you can accurately map out any set of planetary observations, with all the messy details included.

And I do mean any set. (You can watch the whole thing, but skip to about a minute for the good part.)

Video uploaded by Santiago Ginnobili.

Aha! We have discovered that Homer Simpson is really a complex set of epicycles. But no, that seems to have things exactly backward. Homer Simpson comes from the imagination of Matt Groening. That we happen to be able to describe the character's appearance using a set of epicycles does not give us any insight into why the character looks the way he does.

And yet packing in all the detail can give us the illusion of understanding. You see, that complicated set of Homeric epicycles can also be represented as a Fourier series, a sum of sine functions where each function—with a differing amplitude and frequency—represents one epicycle. That is, we can come up with an equation that describes Homer Simpson, and all we have to do is plug in the right numbers. It is easy to imagine, then, that there is a physical reason for each of us those amplitudes and frequencies. Once we've found a reason for each number, we would seem to have a very satisfying scientific explanation for the existence of Homer.

But we know, of course, that there is none, because Homer is not really built up of epicycles and all the detail we've admitted into the system has led us astray.

Okay, but then how do we discover the truth when the details get in the way? How do we "hide" them so we can see what's underneath? Abstraction is the answer. To see how that works, let's get away from early astronomy and back to the bright, messy sun.

At work, I recently came across a very neat technique used for studying the spectra of solar system bodies. So let's say we want to know more about a comet that's recently swung by the inner solar system. When we point our telescope at it, what do we see?

Not a real spectrum of any comet. Just something I cobbled together.

If we put a spectrograph on the back of the telescope—a device that breaks up light into individual wavelengths like a prism—we get this weird hump with spikes running through it. This graph shows how intense the light is at each wavelength, running from violet on the left to red on the right.

The problem with this graph is that most of what we're seeing is just reflected light from the sun. If we want to know about the comet itself, we have to find a way to eclipse (sorry) the sun's spectrum.

This process is known as continuum-subtraction. The shape of our comet spectrum is determined by two factors: absorption/emission lines (the spikes) and the temperature of the sun (the overall hump). We have to separate those two if we want to get rid of the sun. We start by abstracting away the details—those messy spikes—leaving us with the continuum.

To do that, we need to find the best fit line for the spectrum. As with the Homeric epicycles up above, our mathematical tools are powerful enough to write some equation that perfectly fits the line. Instead of a Fourier series, it would be a polynomial—a sum of powers of x with coefficients. You know some basic polynomials: a line is a first order polynomial, a parabola a second order one. For this, we might need, say, a 384th degree polynomial, but we could do it. And again, we could imagine that each power of x and each coefficient has some physical cause.

But then the details are distracting us again. The truth is that most of the jagged bits of the spectrum originate from (a) atmospheric interference, the heat of the telescope, and other noise, and (b) absorption and emission lines that are superimposed on top of the continuum spectrum. So let's keep our epicycles to a minimum and use a simple second or third order polynomial instead.

My parabola. You can't have it.

With the comet continuum in hand, we can then subtract it from the comet spectrum. Doing that leaves behind only the spikes.

There are bits of code out there to perform this operation. I don't recommend subtracting each bit by hand.

The spikes that rise above the noise (which we can calculate) are the light being emitted and absorbed by the comet at particular wavelengths. The location of each line along the spectrum is a result of what element or compound is interacting with the light. With the continuum removed, we know what the actual flux is and can determine how much of the stuff is on the surface of the comet.

Another neat trick lets us figure out the albedo of the comet—how much it reflects the light of the sun. Its albedo also depends on its composition. (Think about how much brighter the earth's icy poles are than its liquid oceans.) You might think we can figure that out just by pointing our telescope at the comet and seeing if the object is bright, but its brightness results from its proximity to the sun, distance from us, size, and finally albedo. The first two parameters are easy to figure out if we track its orbit; the last two require some disentangling.

Step one is to point the telescope at a star with similar characteristics to the sun—a solar analog—and see what its spectrum looks like in our telescope. We can't point the telescope at the sun itself, because a telescope designed to look at faint comets and stars would be blinded by the sun. We also don't want to just take someone else's spectrum of the sun, because then we're comparing observations from different telescopes in different environments. Best to use the same equipment.

Now, any solar analog we find out there isn't going to be an exact duplicate of the sun. So like before, we're better served by ignoring the details of the star's spectrum and finding a solar continuum. If we compare the solar continuum to the comet continuum, we'll see that even though the comet's light is mostly reflected sunlight, there are differences. Not every bit of light that strikes the comet's surface is reflected back. Some will be absorbed, and this absorption is wavelength-dependent. What we can do is divide the comet continuum by the solar continuum and come up with the fraction of each wavelength of light that is reflected—the albedo as a function of color.

If the albedo is low, then the comet is naturally dark. If it's dark but appears bright in the telescope, it must be very large. Conversely, if the albedo is high, the comet is shiny. A dim but shiny comet must be small. So that's size and albedo worked out, too.

I think that about covers it. With these techniques, we can abstract away the distracting details—the light of the sun, the noise in our instruments—and come away with facts about a ball of ice and rock hurtling through space. A comet is a messy, complicated object and nothing but its orbit is easily reducible to a simple law, but we can nevertheless know more about it by pretending we see less of it.