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Saturday, July 30, 2016

Reading List: Parallax

Hirshfeld, Alan W. Parallax. New York: Dover, [2001] 2013. ISBN 978-0-486-49093-9.
Eppur si muove.” As legend has it, these words were uttered (or muttered) by Galileo after being forced to recant his belief that the Earth revolves around the Sun: “And yet it moves.” The idea of a heliocentric model, as opposed to the Earth being at the center of the universe (geocentric model), was hardly new: Aristarchus of Samos had proposed it in the third century B.C., as a simplification of the prevailing view that the Earth was fixed and all other heavenly bodies revolved around it. This seemed to defy common sense: if the Earth rotated on its axis every day, why weren't there strong winds as the Earth's surface moved through the air? If you threw a rock straight up in the air, why did it come straight down rather than being displaced by the Earth's rotation while in flight? And if the Earth were offset from the center of the universe, why didn't we observe more stars when looking toward it than away?

By Galileo's time, many of these objections had been refuted, in part by his own work on the laws of motion, but the fact remained that there was precisely zero observational evidence that the Earth orbited the Sun. This was to remain the case for more than a century after Galileo, and millennia after Aristarchus, a scientific quest which ultimately provided the first glimpse of the breathtaking scale of the universe.

Hold out your hand at arm's length in front of your face and extend your index finger upward. (No, really, do it.) Now observe the finger with your right eye, then your left eye in succession, each time closing the other. Notice how the finger seems to jump to the right and left as you alternate eyes? That's because your eyes are separated by what is called the interpupillary distance, which is on the order of 6 cm. Each eye sees objects from a different perspective, and nearby objects will shift with respect to distant objects when seen from different eyes. This effect is called parallax, and the brain uses it to reconstruct depth information for nearby objects. Interestingly, predator animals tend to have both eyes on the front of the face with overlapping visual fields to provide depth perception for use in stalking, while prey animals are more likely to have eyes on either side of their heads to allow them to monitor a wider field of view for potential threats: compare a cat and a horse.

Now, if the Earth really orbits the Sun every year, that provides a large baseline which should affect how we see objects in the sky. In particular, when we observe stars from points in the Earth's orbit six months apart, we should see them shift their positions in the sky, since we're viewing them from different locations, just as your finger appeared to shift when viewed from different eyes. And since the baseline is enormously larger (although in the times of Aristarchus and even Galileo, its absolute magnitude was not known), even distant objects should be observed to shift over the year. Further, nearby stars should shift more than distant stars, so remote stars could be used as a reference for measuring the apparent shift of those closest to the Sun. This was the concept of stellar parallax.

Unfortunately for advocates of the heliocentric model, nobody had been able to observe stellar parallax. From the time of Aristarchus to Galileo, careful observers of the sky found the positions of the stars as fixed in the sky as if they were painted on a distant crystal sphere as imagined by the ancients, with the Earth at the center. Proponents of the heliocentric model argued that the failure to observe parallax was simply due to the stars being much too remote. When you're observing a distant mountain range, you won't notice any difference when you look at it with your right and left eye: it's just too far away. Perhaps the parallax of stars was beyond our ability to observe, even with so long a baseline as the Earth's distance from the Sun. Or, as others argued, maybe it didn't move.

But, pioneered by Galileo himself, our ability to observe was about to take an enormous leap. Since antiquity, all of our measurements of the sky, regardless of how clever our tools, ultimately came down to the human eye. Galileo did not invent the telescope, but he improved what had been used as a “spyglass” for military applications into a powerful tool for exploring the sky. His telescopes, while crude and difficult to use, and having a field of view comparable to looking through a soda straw, revealed mountains and craters on the Moon, the phases of Venus (powerful evidence against the geocentric model), the satellites of Jupiter, and the curious shape of Saturn (his telescope lacked the resolution to identify its apparent “ears” as rings). He even observed Neptune in 1612, when it happened to be close to Jupiter, but he didn't interpret what he had seen as a new planet. Galileo never observed parallax; he never tried, but he suggested astronomers might concentrate on close pairs of stars, one bright and one dim, where, if all stars were of comparable brightness, one might be close and the other distant, from which parallax could be teased out from observation over a year. This was to inform the work of subsequent observers.

Now the challenge was not one of theory, but of instrumentation and observational technique. It was not to be a sprint, but a marathon. Those who sought to measure stellar parallax and failed (sometimes reporting success, only to have their results overturned by subsequent observations) reads like a “Who's Who” of observational astronomy in the telescopic era: Robert Hooke, James Bradley, and William Herschel all tried and failed to observe parallax. Bradley's observations revealed an annual shift in the position of stars, but it affected all stars, not just the nearest. This didn't make any sense unless the stars were all painted on a celestial sphere, and the shift didn't behave as expected from the Earth's motion around the Sun. It turned out to be due to the aberration of light resulting from the motion of the Earth around the Sun and the finite speed of light. It's like when you're running in a rainstorm:

Raindrops keep fallin' in my face,
More and more as I pick up the pace…

Finally, here was proof that “it moves”: there would be no aberration in a geocentric universe. But by Bradley's time in the 1720s, only cranks and crackpots still believed in the geocentric model. The question was, instead, how distant are the stars? The parallax game remained afoot.

It was ultimately a question of instrumentation, but also one of luck. By the 19th century, there was abundant evidence that stars differed enormously in their intrinsic brightness. (We now know that the most luminous stars are more than a billion times more brilliant than the dimmest.) Thus, you couldn't conclude that the brightest stars were the nearest, as astronomers once guessed. Indeed, the distances of the four brightest stars as seen from Earth are, in light years, 8.6, 310, 4.4, and 37. Given that observing the position of a star for parallax is a long-term project and tedious, bear in mind that pioneers on the quest had no idea whether the stars they observed were near or far, nor the distance to the nearest stars they might happen to be lucky enough to choose.

It all came together in the 1830s. Using an instrument called a heliometer, which was essentially a refractor telescope with its lens cut in two with the ability to shift the halves and measure the offset, Friedrich Bessel was able to measure the parallax of the star 61 Cygni by comparison to an adjacent distant star. Shortly thereafter, Wilhelm Struve published the parallax of Vega, and then, just two months later, Thomas Henderson reported the parallax of Alpha Centauri, based upon measurements made earlier at the Cape of Good Hope. Finally, we knew the distances to the nearest stars (although those more distant remained a mystery), and just how empty the universe was.

Let's put some numbers on this, just to appreciate how great was the achievement of the pioneers of parallax. The parallax angle of the closest star system, Alpha Centauri, is 0.755 arc seconds. (The parallax angle is half the shift observed in the position of the star as the Earth orbits the Sun. We use half the shift because it makes the trigonometry to compute the distance easier to understand.) An arc second is 1/3600 of a degree, and there are 360 degrees in a circle, so it's 1/1,296,000 of a full circle.

Now let's work out the distance to Alpha Centauri. We'll work in terms of astronomical units (au), the mean distance between the Earth and Sun. We have a right triangle where we know the distance from the Earth to the Sun and the parallax angle of 0.755 arc seconds. (To get a sense for how tiny an angle this is, it's comparable to the angle subtended by a US quarter dollar coin when viewed from a distance of 6.6 km.) We can compute the distance from the Earth to Alpha Centauri as:

1 au / tan(0.755 / 3600) = 273198 au = 4.32 light years

Parallax is used to define the parsec (pc), the distance at which a star would have a parallax angle of one arc second. A parsec is about 3.26 light years, so the distance to Alpha Centauri is 1.32 parsecs. Star Wars notwithstanding, the parsec, like the light year, is a unit of distance, not time.

Progress in instrumentation has accelerated in recent decades. The Earth is a poor platform from which to make precision observations such as parallax. It's much better to go to space, where there are neither the wobbles of a planet nor its often murky atmosphere. The Hipparcos mission, launched in 1989, measured the parallaxes and proper motions of more than 118,000 stars, with lower resolution data for more than 2.5 million stars. The Gaia mission, launched in 2013 and still underway, has a goal of measuring the position, parallax, and proper motion of more than a billion stars.

It's been a long road, getting from there to here. It took more than 2,000 years from the time Aristarchus proposed the heliocentric solar system until we had direct observational evidence that eppur si muove. Within a few years, we will have in hand direct measurements of the distances to a billion stars. And, some day, we'll visit them.

I originally read this book in December 2003. It was a delight to revisit.

Posted at July 30, 2016 21:39