Tuesday, May 23, 2017

Rungs All the Way Down

The last lab we run in Astronomy 101 has students simulate observations of distant galaxies and then do some analysis in Excel to discover Hubble's law. By the end, students come up with a rough estimate for the age of the universe. But as I remarked elsewhere, my students seemed more in awe of Excel's tools than in discovering the origin of space, time, and all of existence (including Excel).

I don't want to get too philosophical about why this happened (because the truth is they were probably just bored and wanted the whole thing to be over with) but I suspect we are kind of spoiled nowadays for awesome science news. Everybody knows the universe began with a big bang billions of years ago, and it's difficult to transport people back to a time when such a fact was remarkable.

Yet the discovery of Hubble's law at the end of 1920s represented the culmination of an incredible project going back millennia, one that eventually paved the way for physical cosmology—the concrete study of the structure, origin, and fate of the universe.

What is that project? Figuring out how far away things are. I know, that sounds tremendously dull, but it speaks to something potent about science: the capacity to get answers to questions you didn't ask. The (seemingly) mundane task of finding ever more accurate and applicable ways to measure distance led to an undeniable empirical fact about the origin of the universe without anyone specifically asking deep cosmological questions. That science can do this is remarkable because you're very likely to find the answer you're looking for whenever you ask a specific question. If you stumble upon a totally surprising answer to a question you weren't asking, there's a much better chance that you're not just fooling yourself. (In fact, Hubble was never entirely sold on the significance of his eponymous law.)

So how did this all come about? Well, first I'll give you the punch line. In the 1920s, Edwin Hubble used the gigantic 100" reflector at Mount Wilson Observatory to measure the distance to several far off galaxies. Then, to help build a map of the local universe, he combined this data with spectra of those galaxies collected by Vesto Slipher and Milton Humason. Due to the Doppler effect, the spectrum of a galaxy shifts if it is moving relative to the observer. Hubble discovered that the farther away a galaxy is, the faster it's moving away from us. (Its spectrum is "redshifted" toward longer wavelengths.)

Credit: Edwin Hubble
By a remarkable coincidence, the correct interpretation of this astonishing discovery had already been found by the physicist Georges LemaƮtre. Using Einstein's general relativity, Lemaitre showed that under the influence of mass, the fabric of spacetime itself could expand outward from a "primeval atom" to the entire universe we see today. If you reverse the recession of galaxies Hubble discovered, you can figure out how long spacetime would have to be stretching out to match the observed distances of galaxies today. That length of time is the age of the universe. Furthermore, if the relationship between redshift and distance really holds up, then measuring an object's redshift tells you how far away it is.

But wait a second. That sounds kind of circular, because you needed to know the distances to those galaxies to find this relationship in the first place. How can we possibly know that Hubble's law is accurate as a distance measure if it relies on a distance measure, and why would you need another one anyway? Those are good questions, but you should have been asking them a long time ago. You see, when you're using Hubble's law to find distance, you're hanging from one of the highest rungs on the cosmic distance ladder, and we've been climbing this ladder for thousands of years.

So let's back up for a moment. I completely glossed over how Hubble determined the distances to these galaxies in the first place. Distance is a tricky thing in astronomy because (until very recently) we couldn't go anywhere astronomical. Instead, we are presented with a celestial sphere that might as well be infinitely far away. The objects on this sphere reveal only two pieces of information: brightness and position. From that we must infer distance. Broadly speaking, brightness and position give us two methods for finding distance: standard candles and geometry, respectively.

Brightness by itself is deceptive because if you don't know beforehand what you're looking at, you can't tell if an object is bright because it's (a) nearby or (b) intrinsically very luminous. Finding a standard candle lets you disentangle luminosity from distance so that brightness encodes distance alone. Here's how that works.

To map his galaxies, Hubble performed careful photometry on a class of stars known as Cepheid variables. Cepheid variables aren't exotic stars made from cepheionic matter; they're just a stage in the lifecycle of massive stars. Cepheids are "variable" because they are dying and unstable, causing them to periodically expand and contract. We observe these death throes as a cycle of brightening and dimming.

In the early 1900s, before we knew the astrophysical details, astronomer Henrietta Leavitt analyzed the brightness over time of thousands of Cepheid variables in the Small Magellanic Cloud (SMC). Because this cloud is distinct from other regions in the sky, Leavitt assumed all the stars are roughly the same distance from Earth. Therefore, any difference in brightness between stars is due to differences in intrinsic luminosity. Using that assumption, she discovered that some Cepheids are (on average, at peak) brighter than others, and that the period of their variability scales with their brightness—the brighter a Cepheid, the longer its cycle.

Thus, measuring the period is a proxy for measuring the luminosity. This was astronomy's first standard candle. Because the period tells you how bright the star is supposed to be, if you see a Cepheid in Andromeda with the same period as a Cepheid in the SMC, you know that any difference in brightness is due to distance alone.

If you notice, by itself the standard candle method only tells you relative distances. You can calibrate your Cepheids with those in the SMC, but if you don't know how far away the SMC is, then your distances are just in multiples of the SMC distance, whatever that is. The upshot is you've only climbed down one rung of the cosmic distance ladder. The ladder ends when you can calibrate a cosmic distance with a terrestrial distance.

Standard candles have another built in limitation. Light intensity falls off with the square of distance, so a standard candle that is 10 times farther away is 100 times dimmer. This is why Hubble needed a gigantic 100" telescope. Without it, he could not resolve individual stars in distant galaxies. If a standard candle is too faint to be picked out, you can't do the precise photometry needed to compare it to a reference candle. So there are many rungs on the ladder, with higher rungs involving supernovae, clusters, and even whole galaxies.

But let's continue down the ladder. Historically, the next rung down involved geometry. Using geometry to measure distance usually involves some type of parallax—that is, observing the change in position of a nearby object relative to more distant objects as your perspective changes. We all intuitively know how this works just by looking out the window of a moving car. Utility poles by the side of the road zoom by; cows in a meadow fall back more slowly; distant mountains appear nearly motionless.

From that alone we see the fundamental limitation of parallax methods. The farther away an object is, the less its apparent position changes. And if it's far enough away, your telescope can't make out any difference in position. In general, parallax methods are only good for relatively nearby stars. But they are a crucial rung on the ladder nevertheless.

After Leavitt's law was discovered, astronomer Ejnar Hertzsprung calibrated the Cepheids in the SMC with ones in our own galaxy. Cepheids are pretty rare (they are a short-lived stage in the lifecycle of massive stars, which are themselves uncommon) so there aren't many that are close enough to triangulate just by watching their position shift over the course of the year. Instead, he used a method known as statistical parallax.

This method works by looking at a set of Cepheids that are roughly the same brightness scattered around the sky. If they're the same brightness, then they are about the same distance from the sun, which means they all lie on the surface of a sphere with the sun in the middle. The radius of this sphere is the distance to the Cepheids.

We can find that radius by looking at the motion of these stars. Stars move across the sky because of their own peculiar motion and the motion of the sun relative to the "local rest frame," which is the frame that follows the orbit of nearby stars around the galaxy. Their peculiar motion is basically random, which means you're just as likely to find a star moving parallel to the sun's motion as perpendicular to it.

Now, there are two ways we can measure the motion of stars. One is to look for the Doppler shift in a star's spectrum to see if it's moving toward us or away from us. The other is to look at the star's proper motion, which is its change in angular position on the sky and is perpendicular to its radial velocity. What we want to do is find the proper motion of a star that is perpendicular to the sun's motion. This motion is tangent to the imaginary star circle we've created.

Credit: No one. This graphic simply popped into existence when needed.
We can then pretend that the star is circling the sun and say that the proper motion is that star's angular speed around the sun. Angular speed can be converted to actual tangential speed by multiplying by the radius. That is, the larger the radius of a circle, the faster an object has to be moving to complete the circuit in a given time. Conversely, if we know the tangential speed, we know the radius—the distance to the star.

But we have no way of independently measuring the tangential speed, because the Doppler shift only measures radial speed. Here's where the statistical part of the statistical parallax method comes in. Because we've assembled a large collection of randomly moving stars, we can just guess that the average radial velocity of a star is the same as the average tangential velocity. We find the average radial velocity using the Doppler effect (being sure to subtract out the component of the sun's motion parallel to the radial motion). Then you set the tangential velocity of your star equal to that average radial velocity, divide by its angular speed, and you've got the distance.

The units for radial velocity are going to be something like km/s, which means we have calibrated a cosmic distance to a terrestrial distance and seemed to have reached the end of the ladder. But the truth is the statistical parallax method has other distance measures baked into it, which means we've really just jumped back down to solid ground, skipping several rungs. In particular, finding the true "solar motion" of the sun requires that you already know some distances.

The real way back to Earth involves measuring the change in apparent position of a very nearby star over the course of a year as the Earth orbits the sun. Finding that change gives you the distance to the star relative to the astronomical unit, which is how far the Earth is from the sun. To measure the astronomical unit, astronomers in the 18th century measured the different durations of the transit of Venus from different positions on Earth. Those timing variations corresponded to changes in the position of Venus across the face of the sun. This gave astronomers the distance to Venus (and all other solar system distances, including the AU) in terms of the size of the Earth. To measure the size of the Earth, ancient Greek smart guy Eratosthenes watched how the lengths of shadows changed as latitude changed, which told him how curved the Earth was and consequently what its circumference was.

I've mostly presented the cosmic distance ladder as being a steady climb from the Earth all the way to the origin of the universe. But in reality it looks more like a game of Chutes and Ladders. I've tried to hint at the fact that there are many more methods involved, each trying to make up for some deficiency in another. Two different methods will operate on different scales but overlap slightly. Where they overlap, you can jump from one rung to the next by calibrating one to the other. Jump enough rungs, and you eventually find yourself at the beginning of everything.

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