Double Vision: How We Learned the Distance to the Stars

By Brian Tung

NGC 4414
NGC 4414
Source: NASA

I was, as I've mentioned in other places, away from astronomy for a long period of time, from about the ages of 10 to 30. I don't mean to say that I was ignorant of astronomy, or even that I had no idea what advances NASA was making, for example. But for those 20 years I didn't much care if it was cloudy on a moonless night, or if atmospheric turbulence made the planets twinkle, or if the sky was studded with a couple of dozen or a couple of thousand stars.

Then, in 1994, my wife bought me a telescope for Christmas, a 4-1/2-inch reflector from Orion. That was a decent step up from the 60 mm department-store scope so often vilified by serious amateurs. You might imagine that this gift started me on the way back to active observing, but it didn't.

Two and three years later, the two great comets of the late 20th century, Hyakutake and Hale-Bopp, made their prominent appearances. Hale-Bopp in particular was well-placed for evening viewing, and viewed through the Orion reflector it was impressive, even in light-polluted Santa Monica. Again, you might imagine that this dramatic event would kick me in the pants, astronomically speaking, but it didn't.

When I finally realized that I wanted to, needed to get back into the astronomical swing of things, I wasn't seated at a telescope. It wasn't even nighttime. Instead, I was sitting and eating a burger in a diner, reading a magazine. The time was late 1997, the magazine was Astronomy, and the article--

But now I'm getting ahead of myself. I should start at the beginning. . .

The first person we know of who estimated the distance to a star was Aristarchus, who lived between 310 and 230 B.C. By tracing the shape of the earth's shadow during lunar eclipses, he determined that the earth was about 3 times larger than the moon, which is about right. He also determined, by measuring some angles, that the distance to the sun was at least 20 times the distance to the moon. He concluded that the sun must then be at least 7 times larger in diameter than the earth. This was a drastic understatement, but let's remember he was working without a telescope or precision equipment.

Before Aristarchus died, another Greek, Eratosthenes, determined that the earth itself was about 13,000 km across, meaning that the sun must be at least something like 90,000 km in diameter. Simple trigonometry, based on the apparent size of the sun, suggested that the sun should therefore be about 10 million km away, a monstrous distance.

As it didn't seem logical to Aristarchus that the huge sun should circle the tiny earth, he proposed that the sun was the center of the universe, and not the earth, which he believed revolved around the sun like the other planets.

The greatest of the Greek scientists, Archimedes, rejected Aristarchus' idea almost immediately. His objections were based on common sense observation: after all, it's patently obvious to even the greatest simpleton that the sun revolves around the earth and not the other way. Further, he argued, if the earth were moving, wouldn't we be able to feel it move, especially over such a large distance as 10 million km? And aside from a few volcanos and other seismic events, the earth doesn't make a constant sound of motion such as a horse does when you ride on it.

Finally, if the earth were really moving around the sun, then we ought to be able to see the stars move in little circles throughout the year (you can see this effect by moving your head back and forth and watching the relative apparent positions of, say, your window frame and distant buildings). Since the stars plainly don't appear to move that way, the earth simply couldn't be moving around the sun. The stars are very far away, to be sure, but the principle should be the same and Archimedes was convinced that some kind of apparent motion should result if the earth really were moving. He was right, of course -- but he didn't realize that the stars are so far away that they don't appear to move at all.

Much is made of the Greek world's affinity for free inquiry, but it's often the case that inquiry is most free when it doesn't contradict the obvious facts of life, and Aristarchus' case was no exception. Few, if any, Greek scientists would ever take Aristarchus' idea seriously, at least publicly.

The stars are so very far away, you see. Astronomy texts try to convey the enormity of the distances involved by using the image of the sun as a beach ball, the earth as a walnut a football field away, and so on. It's all to little avail, as far as I'm concerned. The distances are simply too huge for us to fathom; these kinds of distances can only be comprehended as mathematical abstractions.

But let's suppose we try, following Aristarchus, for a closer target -- our own sun. Once the heliocentric solar system was installed in the scientific consciousness, we were able to determine the relative distances between the planets. The distance from the sun to the earth is defined as one astronomical unit (AU). The distances from the sun to Mercury and Venus are easy to determine, and the Greeks could have done so had they accepted Aristarchus' heliocentric model of the solar system. Because these two planets are closer to the sun than the earth, they can only appear, to an earth-based observer, to be so far away from the sun. (See Figure 1)

orbits of venus and the earth

Figure 1. Orbits of Venus and the earth.

The farthest that Venus ever gets from the sun, in terms of angular distance, is about 46 degrees. (For that reason, we never see Venus at midnight.) Using trigonometry, we can determine that the sun-Venus distance is related to the sun-earth distance by the formula:

SV = SE sin 46

which yields a sun-Venus distance about 0.72 AU, which is just about right.

A similar method works for the sun-Mercury distance, too. The other planets, however, follow more distant orbits than the earth, and we need another method for them. For relative distances, we can use Kepler's third law of planetary motion. That law states that the period T of a planet's orbit, expressed in years, is related to its distance d from the sun, expressed in AUs, according to the formula:

T^2 = d^3

Mars, for example, takes about 2 years to orbit the sun. Plugging that into our forumla gives us a d of about 1.6, and Mars is in fact about 1.6 AU from the sun -- its orbit is noticeably elliptical, so its distance varies from month to month.

But what about the actual the length of AUs measured in units we know, such as miles or furlongs or centimeters? The Italian astronomer Giovanni Cassini, for whom the division in Saturn's rings is named, reasoned that if Mars were observed from two different places on earth at precisely the same time it would appear to be in two different places in the sky. (See Figure 2) This effect is called parallax; you can reproduce this effect by holding your finger close to your face, looking at it with one eye, and then the other, and watching it appear to change positions.

parallax of Mars

Figure 2. Parallax of Mars.

By knowing the distance between two vantage points and the apparent difference in position, Cassini was able to estimate the distance to Mars. He carried out the experiment in 1672, with the observations made by Cassini in Paris and a French astronomer, Jean Richer, in South America. From these observations Cassini derived an earth-Mars distance of about 83 million km. Since the earth-Mars distance is about 0.6 AU, his estimate for the AU was about 138 million km. That's a little below the now accepted value of about 150 million km -- later measurements of transits of Venus across the face of the sun yielded much more accurate measurements -- but for the first time a sense of the true scale of the solar system was known, and it was much larger than Aristarchus had ever imagined.

What about the stars themselves? If the parallax method worked for Mars, wouldn't it work for the stars, too? The problem for earlier astronomers was that the parallax angle for Mars is already very small; the apparent change in angular position amounts to about 25 seconds of arc -- about 1/150 of a degree, and thus there was no hope of determining the stellar parallaxes of stars further than about 25 AU away with the technology of the time. It was tried and sure enough, it failed, so aside from knowing that the stars were more than 25 AU away, not much more was learned.

However, Mars also exhibits another parallax effect, called retrograde motion. It doesn't appear to move across the sky in a uniform direction as the nights pass, but instead sometimes moves forward, sometimes backward. Generally speaking, it progresses from west to east, but as the earth laps it every 26 months or so, it occasionally appears to move from east to west. This is a result of viewing Mars from the moving platform of the earth, and it occurred to astronomers that the same effect could be used to measure stellar parallaxes. Instead of observing the stars from opposite sides of the earth, in other words, we could observe the stars from opposite sides of the earth's orbit. This expanded the base of operations from the approximately 10,000 km of the earth's diameter to the more than 300 million km of earth's orbit, meaning that the distances that could be measured by means of parallax also expanded by a factor of 30,000.

The distances to the outer planets can also be worked out in this manner, but the computations get more complicated because the other planets are moving. On the other hand, over the course of a year most stars don't move noticeably, and the ones that do move do so very steadily, in straight lines, unlike the planets, whose motions are relatively complex.

Observation concentrated on the stars that do move, because it stood to reason that they were the closest. In a crowded room, the people closest to you seem to pass through your field of vision relatively quickly, whereas someone far away will take longer. You can also tell that someone is closer because of their apparent size, and because everything's still close enough for your binocular vision to give you a true three-dimensional picture of the room.

Stars, on the other hand, vary tremendously in size, and they're so enormously far away that even the closest ones are point sources as far as we're concerned. Our two eyes, providing a measly 10 cm or so as a base for parallax, can't come close to yielding a three-dimensional picture of the stellar neighborhood. So astronomers used apparent motion -- in the case of stars, called proper motion -- as a guide in guessing which stars might be the closest and therefore exhibit the greatest amount of parallax. (This motion is termed proper not because it's "correct," but because it's "proper" to the star itself, and not a result of the earth rotating or revolving around the sun.)

From which part of the sky should a star be chosen for observation? As the earth orbits the sun, all the stars should exhibit an east-west motion due to parallax. This motion, unfortunately, is difficult to detect because the earth rotates, meaning that all stars move from east to west throughout the night. In order to detect an arc second of parallax, one would need a clock accurate to 1/15 of a second, the amount of time it takes a star to move 1 second of arc. In the 18th century, no such clock could be built.

What about other kinds of motion? Parallax actually causes all stars to trace out circles, with both north-south and east-west motion, but only far above the earth, at the ecliptic pole, do those circles actually appear circular. Everywhere else in the sky, the circles are foreshortened along the north-south axis and appear like ellipses stretched out east-west. Thus, stars at the ecliptic pole show the greatest north-south motion due to parallax. North-south motion is easier to discover than east-west motion, because detecting it only requires you to check where it passes overhead, without worrying about accurate timekeeping. In 1728, the English amateur astronomer James Bradley attempted to detect the north-south parallax of the star Eltanin (gamma Draconis) as it passed each day over a specially mounted zenith telescope. If it crossed the field of view of the telescope at a slightly different position as the months passed, that would have to be the parallax and he could thereby determine the distance to that star.

Sure enough, Eltanin passed overhead at a slightly different point each day, moving slowly but steadily southward. Theoretically, a star far from the ecliptic should reach the northernmost extent of its parallactic path when the sun is most precisely between the earth and that star in its orbit around the sun -- when the star is at conjunction. On that date, that star should pass directly overhead at around noon local time. Bradley watched patiently each day as Eltanin passed further and further south, and kept right on progressing south, even as the date of conjunction passed!

What Bradley had discovered was not stellar parallax, but what is known as the aberration of light, and it was some time before he arrived at an adequate explanation of it. I'll explain it by making analogy to raindrops on an automobile windshield. If we suppose that there is no wind, and the car is stationary, then the drops appear to the observer, correctly, to be falling from straight above. If, however, we start the car in motion, then the raindrops ahead of us appear to be moving toward us, and those behind us appear to be moving away from us. The end result is that the raindrops appear to be coming from a point directly ahead of where they "ought" to be coming from.

Now, the speed of light is many millions of times faster than the speed of raindrops, and the speed of the earth is only many thousands of times faster than an automobile, but even so, the effect is detectable, and is also at least an order of magnitude or so larger than parallax. What's more, whereas the parallax effect is determined by the earth's position relative to the sun, the aberration effect is determined by the earth's motion around the sun, and these two are at right angles to each other. This explains why Eltanin continued southward past the time that would have been predicted from parallax alone. (See Figure 3.)

effects of parallax and the aberration of light

Figure 3. Effects of parallax and the aberration of light.

As you can see, the aberration effect totally swamps the parallax effect, and that's true even for the nearest of stars. As you observe stars that are more and more distant, the aberration effect remains constant, but the parallax effect grows smaller in proportion to the increasing distance. In fact, it's only because the aberration effect is constant that parallax can be detected at all -- by determining the relative change in position that a star makes in contrast to its neighbors. It's no knock on Bradley that he was unable to detect Eltanin's parallax with the technology of his time, because Eltanin is nearly 150 light years away.

Just a decade after Bradley's discovery, the German astronomer and mathematician Friedrich Bessel observed a parallax of 1/3-arc second on the star then known as 61 Cygni. This means that as the earth moves around the sun in its orbit, 61 Cygni appears to trace out a little oval that is 2/3 of an arc second in width. By convention, the parallax is defined as half of this width, just as the distance from the earth to the sun is half of the base from which the parallax is measured.

How far away, then, is 61 Cygni? We can express this in terms of a parsec, a distance is equal to 206,265 AU (this number, which has an almost mystical significance to astrometrists, is equal to the number of arc seconds in 1 radian). Distance is inversely proportional to parallax, so 61 Cygni must be 3 parsecs away. A parsec is equal to about 3.26 light years, so Bessel estimated the star to be almost 10 light years away. That's a bit of an overestimate, but pretty good for a first try, and in commemoration of this feat 61 Cygni is now informally known as Bessel's Star.

(Actually, as is so often in the history of science, Bessel wasn't quite the first to determine a stellar parallax. Working a quarter of a world away, the South African astronomer Thomas Henderson determined the parallax of Alpha Centauri, which is the closest star visible to the unaided eye, outside of our own sun, and he did it before Bessel had arrived at a figure for his star. Possibly because of his remoteness, however, he didn't publish his result until months after Bessel, and it's for that reason that Bessel gets the credit, and not Henderson.)

Determining parallaxes is not an easy task. First of all, the image of a star, even through perfect optics, is not a point of light, but rather a small disc called the Airy disc. This disc gets smaller as the optics get larger, but even for a mirror 1 m in diameter the disc is still about 1/8-arc second across, and Bessel certainly didn't have access to anything that large for the purposes of determining stellar parallax.

Moreover, the atmosphere can wreak havoc with the image of a star, distorting it and moving its apparent position back and forth. The upshot is that the uncertainty of a star's position on any given night may be many times greater than the expected parallax, and it requires careful observations taken over a period of months to obtain even a rudimentary parallax measurement. For this reason, using parallax to determine the distances to the stars was a method limited to stars within about 10 parsecs even well into the 1980s.

Then, in 1989, the Hipparcos (High-Precision Parallax Collecting Satellite) satellite was launched into earth orbit in order to avoid atmospheric effects. It almost didn't make it. After a perfect launch, one of its booster rockets failed to fire properly. Instead of settling into a nearly circular orbit as planned, the satellite revolved around the earth in a severely elliptical orbit which, at its closest approach, brought it within about 400 km of the earth's surface, a height at which the atmosphere may begin to affect it. At first it was feared that all was lost, but despite the near catastrophe Hipparcos functioned quite well, and returned four years of high-precision data on 120,000-odd stars and another set of lower-precision data (but still very high by human standards) on about a million more.

It took four years more to digest the data. When it was done, accurate parallaxes had been determined for most of those 120,000 stars, reaching out to 100 or 200 parsecs. The results are available in a 16-volume report. Volumes 5 through 9 are the so-called Hipparcos catalogue, giving detailed astrometric and photometric data for the 120,000 stars. Volumes 14 through 16 are the three-volume Millennium Star Atlas, a $250 or so "atlas to end all atlases." And finally, the whole story was written up in summary form in an article for the December 1997 issue of Astronomy, which is where I began this story.

My return to astronomy really began with that article, and below are the fruits of that return. The first thought I had upon reading this article was how nice it would be to have a three-dimensional map of the stars. I had just finished a prototype of a program to generate single-image random dot steroegrams, or SIRDS, which are the ancestors to those funky three-dimensional posters that you see in the malls. It occurred to me that I might easily use the same principles to produce three-dimensional stellar pictures.

It turns out that the entire Hipparcos catalogue is available on the Web. It takes a high-speed connection to download it within a reasonable period of time, but once you have it, even a computer a couple of years old can manipulate it reasonably easily. I did end up writing the program to generate the pictures, and below are some results. The inlined images are regular 2-D pictures; clicking on them will bring up the 3-D stereograms. At the end is a link to a form which allows you to generate your own stereograms of whatever portion of the sky you wish.

(Note: I get much better results when I view these pictures with a "helper" application than I do using Netscape to view them. Your mileage, as always, may vary.)

The Big Dipper

the big dipper

Our first example is the Big Dipper, the brightest part of Ursa Major. It consists of seven stars in the shape of a dipper, or plough; its name in the U.K. is "the Plough." Normally, constellations (or "asterisms") are chance collections of stars; that is, they are not physically associated, but are merely along the same line of sight, just as putting your thumb in front of the moon does not make it physically associated with the moon.

Not so with the Big Dipper. Out of the seven stars that make up the Dipper, the central five are in a loose association, some 80 to 90 light years away. Only the outer two, Benetnasch (Eta Ursae Majoris) in the east (upper left here), and Dubhe (Alpha Ursae Majoris) in the west (lower right here), are not part of this group.

The stereogram here (click on the above image to retrieve it) illustrates this fact. If you fuse the left image with your left eye with the right image with your right eye, you'll see a stereo image that shows the central five stars of the dipper hanging closer than the outer two.

The Pleiades

the Pleiades.gif

The Pleiades are perhaps the most famous open star cluster. Open clusters are groupings of stars that are newly born. Typically, we see the brightest of them, which are hot blue stars destined to live only a few tens to hundreds of millions of years. Occasionally some of them have already passed beyond their normal lifetime and are swelling into the red supergiant phase, after which they will blow themselves up as violent supernovae.

Also called the Seven Sisters, only six of the Pleiades are easily seen by the average naked eye, though some sharp-eyed observers can see more than a dozen. It helps to be experienced enough to know where to look for the dimmer ones. While pictures of the Pleiades show some nebulosity enveloping the Pleiades (once thought to be the cloud from which they were born, but now thought by some to be a chance cloud in their path), it's actually quite difficult to see, and most telescopes will show only the stars themselves. To see if you might be seeing the nebulosity, check with the Hyades, to the southeast of the Pleiades, in the direction of Aldebaran in Taurus the Bull. If you see it there, too, then you're not really seeing it; the Hyades are not swathed in nebulae.

In this view here (click on the above image to retrieve the stereogram), we have moved by 95 parsecs, or about 310 light years, in the direction of the Pleiades, in order to show more detail in their spatial placement. The brightest star, near the middle, is still Alcyone (Eta Tauri). Estimates place the population of the Pleiades at over 100 stars.

The Hyades from Afar

the Hyades

If it weren't for the Pleiades, the Hyades would be the most famous open cluster in the sky. They mark Taurus the Bull's snorting visage, and are marked in the sky by the Bull's red eye, Aldebaran. Aldebaran only looks as if it's part of the Hyades; in reality, it is a foreground star only a mere 20 parsecs (about 65 light-years) away.

But in this view, we haven't shown the Hyades as they appear from earth. Instead, we've moved about 15 parsecs (about 50 light-years) in the direction of Orion's shield shoulder. In addition, we are looking sideways, so that east is up and west is down. This allows us to show the separation between the Hyades and Aldebaran when they are looked at obliquely; the Hyades are at the top, and Aldebaran is actually the very bright star at the bottom. In between, actually, are the more distant Pleiades.

The Coma Berenices (Star) Cluster

the Coma Berenices Cluster

The Coma Berenices Star Cluster has to be distinguished from its larger but somewhat lesser-known neighbor, the Coma Berenices Galaxy Cluster. The latter is a target only for larger telescopes, but the Star Cluster can be seen by the naked eye under reasonably dark skies. Binoculars will display it in all its splendor, the brightest stars looking like jewels in the sky.

The cluster is supposed to represent a head of hair. Some people consider the resemblance obvious; others consider it a sign of obvious mental disturbance. The view here (click on the above image to retrieve the stereogram) reveals how the cluster would look if we were to approach it by 35 parsecs, or about 115 light years. This view reveals most of the bright stars in the cluster to form a true open cluster, rather than a chance alignment.

Insert Favorite Asterism Here

I've set up a page whereon you can make your own stereogram. The images look just like the ones here, except that you can pick your own coordinates, your own magnification, and select more or fewer stars based on apparent visual magnitude. You can even move out into space! Give it a try here.


Brian Tung is a computer scientist by day and avid amateur astronomer by night. He is an active member of the Los Angeles Astronomical Society and runs his own astronomy Web site.

Adapted from Astronomical Games, October 1999.