How Do We Know What We Know?

musings on scientific knowledge

The Evils of SI Units

The post title is kind of a misnomer: I actually think SI units are a net positive. No reason to spend a lot of time doing complex unit conversions. That being said…

I think the introduction of SI units in classes–and the lazy way lots of teachers, myself included, do examples–hurts the way students learn science. In particular, it tends to make people think that units don’t matter.

Now, the particular unit you choose for one dimension–such as choosing feet or meters for length–doesn’t matter a great deal. I am the same height if you say I am 5.50 feet tall or 1.67 m tall. But that dimension is really important: it’s nonsensical to say I am 1.67, without reference to a system for measuring lengths, or to use an inappropriate unit (I am not 100 Watts tall). The problem is that you can usually get the right answer in SI without keeping track of your units step to step, and so beginning students often think the dimension as well as the unit is unimportant. This problem is particularly noticeable in astronomy because we use so many idiosyncratic units (a problem in itself, but anyway…).

In some sense, if you’re using SI units, the units really aren’t that important and, if we were more sensible, nobody should need the kind of facility most astronomers have with unit conversions1. As long as you know what dimensions you need to have, keeping track step to step is one of those things like simple algebraic manipulations that–given practice–you can usually do in your head, going back only if you can tell at the end that you’ve screwed up. But on the other hand, sometimes it’s the units themselves that tell you you’ve screwed up! In some classes I’ve taught, students get tangled up with G, the gravitational constant, and g, the acceleration due to gravity on the surface of the Earth. They have different units, so you’d be able to tell if you had the wrong result if you were careful doing the calculation–the units are important not just because they’re part of reality, but because they help you understand the problem you’re doing.

How do we solve this problem? I think a bit of dimensional analysis is one option, especially in physics classes. It helps illustrate that you can sometimes figure out how do to a problem just by the dimensions involved. Also, instructors need to be really careful in introductory classes not to drop the units in intermediate steps. It’s a pain, but it’s really important for understanding.

And we keep the SI units. What can I say, powers of ten make me happy.

1: No unit is going to have easily comparable values when we deal with scales from optical wavelengths (~10-6 m) to the distance between galaxies (~1022 m). Why do we insist on sticking with the centimeter when everybody else is using the meter? Are we really that perverse??2 back

2: Yes. Also, I think we’re stuck with arcseconds.back

The fear is unfounded

The start-up had been preceded by some well-publicized hysteria on the fringes, with alarmists worrying that the L.H.C. would create a black hole that could swallow the earth. (The fear is unfounded.) There was also a cern subplot in Dan Brown’s Angels and Demons, in which Illuminati steal anti-matter from the L.H.C. in order to evaporate the Vatican. (Also not a concern—it would take an impossible amount of time and energy to produce enough anti-matter to make a bomb.) source

I have to say, it’s so nice to read a news article that actually just says when fringe opinions are fringe opinions. This gets into one of my things about the language of science, which is that we don’t like to say things are impossible: that’s like asking your next experiment to prove you wrong. We know we don’t know everything about the universe; finding the impossible stuff is the point. But that tends to give lay observers the sense that things are more probable than they are, because in other situations people round “deeply unlikely” down into “not at all possible,” and many scientists won’t.

Take the thing about the LHC destroying the universe. Before it was turned on, could we say with absolute certainty that it wouldn’t? No, because we can’t say anything with absolute certainty. But were any physicists involved with the experiment saying their last goodbyes to family members? No. (Well, perhaps jokingly.)

I like the phrase “the fear is unfounded.” It captures the right sense: not that the thing is impossible, but that it’s sufficiently unlikely that one need not worry about it.

Hubble, nebulae, galaxies, and our place in the universe

One hundred years ago, most people thought that the galaxy and the universe were the same thing.

It’s weird to think about, looking at photos like this one of galaxy M101 from the Astronomy Picture of the Day. But there are a couple of things to realize. One is that most of us have been trained to see these images as galaxies–it seems perfectly obvious, because we’ve seen lots of them with captions about being this or that galaxy and because we already accept the notion that there are many galaxies in the universe. If you think that it’s very unlikely there are other galaxies–if you are skeptical of things being so far away, when we’re already a trillion times as far away from the center of the galaxy as we are from the Sun, when the Sun itself is the about same distance as forty thousand trips between New York and Los Angeles–then maybe you look at an image like this and see shreds of gas being pulled into a newly-formed star system. But using only images of what were then called nebulae, Edwin Hubble managed to show that some of them were in fact other galaxies.

Hubble, for whom the telescope was named, presented some evidence in his 1917 Ph.D. thesis on faint nebulae that supported the nebulae he observed being outside our own galaxy, not part of it. First of all, he saw most of his nebulae in directions that suggested they were above or below the disk of the galaxy, when you’d expect to see them concentrated in the disk itself like the stars are. He says, “Suppose them to be extra-sidereal and perhaps we see clusters of galaxies; suppose them within our system, their nature becomes a mystery.” (p. 7) He also attempted to measure rotational speeds and relate them to densities and masses, and under a number of assumptions (some quite bizarre to modern eyes) found they made more sense if you assumed the nebulae were very far away, about a million light years. So Hubble was clearly on the side of the nebulae being galaxies.

There was a pretty famous debate in 1920 between Harlow Shapley and Heber Curtis, arguing about the size of the Milky Way and the existence or not of other galaxies. This was settled in 1923, when Hubble got the first distances to other galaxies, a process that’s worth talking about in itself.

One of the main limits in cosmology is that you can’t go out and measure distance directly for objects that are far away. All we can get is the characteristics of the light coming from these objects: the amount of light (the flux), the color (spectrum) of the light, changes in both those things with time. Hubble used both the amount of light and the change of light with time to figure out the distances to some of his faint nebulae.

We know the relationship between flux and luminosity. Flux is how much energy from light hits an area of a given size in a given amount of time: for example, a 1 square meter solar panel just outside the atmosphere of the Earth pointed directly at the Sun is hit by about 1400 joules of energy every second, so we’d say the Sun’s brightness is 1400 Joules/second/meter2, or 1400 Watts/meter2. Luminosity is the total amount of energy an object gives off in a given amount of time: in the Sun’s case, about 4×1026 Watts. Flux goes down as distance from the bright object goes up. If you’ve ever pointed a flashlight at a wall and walked forwards or backwards, you’ve probably got a sense of this. When you’re close to the wall, the spot is small and bright, so there’s a lot of flux; when you’re farther away, the light spreads out and the spot is bigger but dimmer. The total amount of light coming off the wall is the same, of course, because it’s all the light coming out of the flashlight, which doesn’t change; in this analogy this total amount of light is the luminosity of the flashlight. Galaxies give off light in all directions, and we can’t see all of it–so it’s like we’re a tiny speck on the wall, seeing only whatever light from the flashlight hits us, which will lessen as the flashlight moves away. The light from the flashlight spreads out on the wall, and the total amount of it we’re getting (the flux times the area) is basically the size of our speck divided by the size of the giant flashlight-spot on the wall, times the luminosity of the flashlight itself. You can describe the whole system with three quantities: the luminosity of the light bulb, the flux on our speck on the wall, and the distance between them.1

Here’s the problem, then: how do you know how bright the flashlight is to start out with? You definitely don’t know the distance, so you have to know the other two quantities in that equation. The answer is that there are a few things we know the intrinsic brightness of, and the best one is Cepheid variables, a kind of pulsating star2. There’s a relationship between the time it takes to do one pulse and the luminosity of the star, which was worked out by Henrietta Swan Leavitt and which is a topic I’m going to leave for another post. Anyway, Hubble managed to find some Cepheid variables in these distant galaxies, and from them derive the distances–vastly farther than even the largest estimates of the size of the Milky Way. This evidence was accepted very quickly by the astronomical community: there were many galaxies in the universe, not just our own, and the universe was far larger than we had realized.

1: I’m ignoring that the wall is flat and the flashlight close–the relationship we use in astronomy usually relies on the diameter of the area of observation being much less than the distance between the observer and the source being observed. We assume that we’re well into the small angle approximation regime, in other words. Hard to work that into the analogy without getting bogged down, though. return

2: Not a pulsar, that’s something different. return

Follow

Get every new post delivered to your Inbox.