Astronomical Games: May 2004

Through a Glass, Lightly

How can curved glass be used to stretch celestial objects before our very eyes?

We draw a magic circle and shut out / everything that doesn't agree / with our secret games. / Each time life breaks the circle, / the games turn grey and ridiculous. / Then we draw a new circle and build a new defense.

—from the movie Through a Glass, Darkly (1961)

THERE SEEMS to be a perception that amateur astronomers are not prone to watching television—in fact, more than that, they find even the thought of watching television to be revolting. I'm here to tell you that that doesn't to be the truth in my case. I actively enjoy watching television. I'll probably watch just about anything, but I particularly enjoy watching sports and detective shows, and one of my favorite detective shows is Columbo, starring Peter Falk.

Columbo is a surprisingly dark show at times, something that it might have inherited from its salty star. Falk has a glass eye—something that he has had ever since he was three, when a cancer was found in his right eye, and it had to be taken out. This didn't prevent him from playing sports as a child—baseball, in particular.

One time, when Falk was playing little league ball, he slid into third base and was called out by the umpire. As any player would in those circumstances, he called the umpire blind—and not in the good way that, say, justice is blind.

But having a glass eye gave Falk an added twist. He reached into his right eye socket, angrily pulled out the glass eye and handed it to the ump. "Here!" he yelled. "I think you might need this!"

Of course, we might ask ourselves how it is we see anything at all. Clearly, our eyes are involved, but what role do they play? The Greek philosopher Plato (427–347 B.C.) thought that the eyes emitted light rays, which then reflected off, say, a rock in front of us, back into our eyes—whereupon we saw the rock.

It's easy with our modern hindsight to laugh at this: If Plato were right, shouldn't we be able to see in the dark without benefit of fire or flashlight? But this requires that we understand darkness as the mere absence of light. If, instead, darkness is something more active—something that absorbs light, for instance—then matters become less clear. It might be that the surroundings are in some way too dark for our light-emitting eyes to work.

Plato thought that it was the lens of the eye that emitted the light. It was the German astronomer Johannes Kepler (1571–1630) who identified the actual functioning of the lens. As he described in his book, Ad Vitellionem Paralipomena (published, incidentally, in 1604, in the midst of his work on planetary orbits), he conducted simple experiments that showed that the lens and the cornea were simple refracting elements that let light through. He also described the mathematics of refraction and explained how lenses work, both in the eye and in spectacles. (The name of the book is Latin for "appendix to Witelo"; Erasmus Witelo was a thirteenth-century Polish scholar who wrote a treatise on optics that became the standard for a few centuries.)

Kepler couldn't, however, figure out the exact functioning of the retina, the layer of cells at the rear of the eye's interior surface, although he did figure out that visual detection was happening there. The details would take three centuries of careful study. But he did work out a high-level understanding of how vision works.

In this conception, we see (let's say) an ant when light strikes the ant and bounces off into our eye. An ant is not like a mirror; it's not shiny. In optical terms, a light ray that strikes the ant's head is reflected in all different directions, in a fan of diverging light rays. Some of these light rays enter our eye, where it is refracted by the eye's lens, so that the rays are now converging instead of continuing to diverge. (To be precise, a considerable amount of the refraction is also performed by the eye's cornea, but for the sake of simplicity, we'll refer to just the lens.) If the eye is focused properly on the ant, those rays converge to a point on the retina, where light-sensitive detectors record the image of the ant's head. (See Figure 1.)

eye focusing on an ant
Figure 1. How the eye focuses on an ant.

Furthermore, only the ant's head will be seen by that part of the retina; another detector will capture light reflected by the ant's tail, and so forth. These detectors convert the light they receive into electrochemical signals that are sent to the brain, where they are processed. At that point, we see the ant.

In order to see the ant clearly, the eye has to be focused properly on it. This means that the lens must refract the light just so, for the light rays from each tiny part of the ant to converge to its own point on the retina.

Why might the light rays converge to a point off the retina? One possibility is that the ant is too close to the eye. In that case, the light rays reflecting off the ant are diverging so steeply that the eye's lens is simply unable to get them to refract them enough. The light rays passing through the lens are converging so gently that they reach the retina before meeting at a point.

Something similar happens if the ant is too far. Here, the light rays entering the eye are diverging too gently, and after the lens refracts them, they are converging too steeply. They therefore meet at a point in front of the retina. In either case, the ant is out of focus.

In order to keep objects in focus as they come closer or move further away, the eye's lens is not attached directly to the eyeball, but is rather connected to it with tiny muscles called the ciliary muscles. These muscles flex and relax the lens so that the light rays converge onto the retina. Our ability to use the ciliary muscles to keep objects at different distances in focus is called accomodation.

As we grow older, our ability to accomodate lessens. For some of us, books at what used to be perfectly ordinary distances, like 10 inches, are a strain to read. We have to hold the book back far enough to allow the eye to converge the light onto our retina. This condition is called hyperopia, or farsightedness. Other people (such as myself) can focus just fine on nearby objects, but cannot focus on distant objects. This is called myopia, or nearsightedness. Nowadays, such conditions can be corrected for with glasses or contact lenses, or even, in some cases, repaired using eye surgery.

When we look at a small notice on a distant bulletin board, there is no way we can read it, because the writing on the notice is too small for us to read at that distance. In terms of what's happening in the eye, the eye may be able to focus on the letters in the notice, but the letters are so small that light from any of them focuses entirely onto the same detector on the retina. As far as the eye can tell, all the letters are indistinguishable and the notice can't be read.

The most straightforward way to solve this problem is just to walk closer to the notice. At some point, we're able to read the notice. The letters themselves are no larger than they were before, but because we are closer to them, light from different letters now converge to different points on the retina. We can now distinguish those letters, and read them.

We can get a sense of what's happening optically by applying one simple rule about lenses: They refract light everywhere but at the center. (This isn't exactly true, but it's close enough for our purposes.) Light passing through the center of the lens does so without being bent in the process.

We can use this rule to figure out where an object will form an image on the retina, provided the eye is properly focused on it. All we have to do is draw a line from the object, through the center of the lens, and onto the retina. That is where the image will be formed.

One thing we can immediately tell from this rule is that the image formed on the retina of any object will be upside-down. If we trace the light rays reflected from the top and bottom of a tree, those rays will cross at the center of the lens. When they reach the retina, the image formed by the top of the tree will be below the image formed by the bottom. The image of the tree is thus upside-down. We don't see the tree as upside-down because the brain is wired to accept an upside-down image.

(In fact, the connection can be rewired. There have been experiments in which the subjects wore special glasses that inverted the image before it entered the eye. The eye in effect recorded a right-side-up image, which the brain had trouble interpreting. The subjects initially had a lot of difficulty getting around, but by the end of the experiment, their brains had rewired themselves to accept the newly oriented image. Of course, they had to rewire themselves all over again once the experiment was concluded.)

The second thing we can tell from this rule is that your eye is equally able to discern objects 1 cm tall at a distance of 1 m, or 2 cm tall at a distance of 2 m, or 5 cm tall at a distance of 5 m, or in general, objects with a height of x at a distance of 100 x, where x is any length. That's because despite being all different sizes, they all form images of the same size on the retina, as you can easily verify.

Another way of expressing this is that they all have the same angular size. As we observed, if you trace the light rays from the two ends of an object through the center of the lens, they cross at that center. The angle of intersection is the angular size of the object. Notice that objects of the same angular size—even if they have different linear or true sizes—all form images of the same size on the retina.

Thus, we can say that when we're too far from the notice to read it, the letters have too small an angular size. We can double the angular size by either doubling the linear size of the letters, or by simply halving our distance to them. At some point, the angular size becomes large enough for the notice to be readable. That point varies from person to person, but generally speaking, the letters have to be at least a few arcminutes in angular size, where each arcminute is just 1/60 of a degree.

Now, is there a way to increase the angular size of an object without either increasing its linear size or decreasing our distance to it?

One place where this would be handy is in inspecting small objects. Consider really small writing, like letters on microfilm. Such letters may have a linear size of only about 10 microns, about a tenth the size of an amoeba. In order to make their angular size be a few arcminutes, we would have to put our eye within about 1 cm of the letters. Just about no one can focus on any object that close. The light rays reflected by the letters are simply diverging too steeply at that distance to be focused onto the retina by the eye's lens. The closest that most people can get is about 10 cm, at which point the angular size is only a few tenths of an arcminute, making the letters unreadable.

For some time now, people have been using glass lenses to aid in looking at small things. Some lenses have been found to be extremely old; the Layard lens, found by the English archaelogist Austen Layard (1817–1894) in the ruins of Babylon (now Iraq) in 1850, dates back to the late 9th century B.C., about the time that Syria and Babylon were taking the lead in glassmaking. It's unclear whether such lenses were used optically or not; the Layard lens is not usable in that way today, but there is some thought that this has resulted from damage caused by compression over time, as well as when the lens was first recovered.

Part of the reason there is any doubt that such old lenses were used in any optical way (as opposed to decorative purposes) is that not any old curved piece of glass will do. The eye's lens works as well as it does because it is specifically curved to bring light rays to a focus, and the same is true of any other glass lens used in spectacles, telescopes, microscopes, and so forth.

These lenses have a characteristic quantity called their focal length. The focal length of a lens is defined as follows: If you pass a parallel bundle of light rays—that is, rays that are neither diverging nor converging—through a lens, those rays will converge to a point called the focal point. The focal length is the distance between the focal point and the lens. Consider a lens with a focal length of 1 cm. In that case, parallel light rays come to a focus 1 cm behind the lens.

It works both ways. If you put the lens 1 cm away from a source of diverging light rays, the lens refracts those rays to make them parallel.

Now, suppose you put the lens just a little less than 1 cm over the minuscule letters—say, about 9 mm. At that distance, the fan of rays being reflected by any point on the letters is diverging just a bit too steeply to be made parallel by the glass lens. When the rays exit the lens, they will still be diverging, but much more gently than before. These new light rays appear to be diverging from a point under the lens, but because the divergence is gentler, that point looks as though its further beneath the lens than the letters really are. Specifically, the rays appear to be diverging from a point 10 cm beneath the lens, rather than 1 cm.

That's not all. If we trace those light rays for the top and bottom of any of those 10-micron-tall letters, we find that they appear now to be coming from letters that are 100 microns tall. (See Figure 2.) And if we put our eye right on top of the glass lens, we will in fact see letters that are 100 microns tall, 10 cm away. They aren't really there—we might call them virtual letters—but they look as though they are, and we can read them as easily as though they were really there.

magnifying glass focusing on letter
Figure 2. The way a magnifying glass works.

You might say that matters haven't really improved. There's no difference in angular size between letters 10 microns tall at a distance of 1 cm, and letters 100 microns tall at a distance of 10 cm. Both of them have an angular size of a few arcminutes.

Ah, but there is a difference. Your eye can focus, just barely, on the virtual letters 10 cm away, whereas they can't focus on the real letters 1 cm away. You can make things even better. If you raise the lens to 9.5 mm above the real letters, the virtual letters will be 200 microns tall at a distance of 20 cm, making it easier still for your eye to focus on them and read them. None of this makes their angular size any larger, but your eye can now focus on them more easily.

And if you put the lens 1 cm (its focal length) above the letters, the virtual letters will now appear to be an infinite distance away, but still with the same angular size of a few arcminutes. If you aren't nearsighted (or, even if you are, if you wear corrective lenses), you shouldn't have any trouble focusing on the virtual letters.

One way to summarize all of this is that the magnifying glass (for that is all the glass lens is) makes it possible for you to observe things as though your eye were just where the magnifying glass is, in terms of angular size, but in such a way that your eye has no trouble at all in focusing. And since the light rays being focused with no trouble by your eye don't really come from anything, but are instead traced back, the image seen by your eye through such a glass is called a virtual image.

So much for observing things both tiny and close up. What about the large and distant?

At first glance, it doesn't seem as though the magnifying glass can help us at all. We see that it's useful for getting a closer perspective on things, so to speak, but we can't make magnifying glasses with focal lengths of millions of light-years to observe galaxies. Even if we could, that would only permit us to see the galaxies from the perspective of the magnifying glass, which we can already do.

It would help if we could make a small copy of the galaxy in mid-air. Then we could hold our magnifying glass to it, and see it as close up as we liked. Unfortunately, there's no simple way of doing that. Or is there?

Remember that when a object has an angular size of 5 degrees, say, that means that if we trace the light rays from either end of the object through the center of the eye's lens, those rays intersect there at an angle of 5 degrees. Because the center of the lens passes light rays through unbent, the object's image on the retina also has an angular size of 5 degrees, as seen from the point of view of the lens.

All this has nothing inherently to do with the retina itself. The image would still be formed in the same place, with the same size; the retina is just there to detect it.

Nor is there anything special about the eye's lens. Any convex lens (that is, one that is curved outward) at all will form an image of an object at some distance provided the object isn't too close to the lens. So long as we're looking at a galaxy (which is effectively infinitely distant), the image will be formed, and the distance between it and the lens is simply the focal length of the lens.

Suppose we place an imaginary observer right where the lens is. We'll name this observer Janus, after the Roman god with two faces, since Janus will be able to look out simultaneously in one direction at the galaxy, and in the opposite direction at the image of the galaxy, formed by the lens. If the lens works the same way as in the eye, the galaxy and its image should have the same angular size to Janus.

To be sure, Janus normally can't see the image, since it is formed by light rays that have already passed through the lens. In order for Janus to see it, light rays from the image would have to somehow come back toward him. Left to their own, they clearly won't do that, but if the image is bright enough, we can put a screen right where it's formed, and then Janus can see it. (In a sense, the retina is just a screen on which the eye's lens projects an image.) At that point, Janus can verify that the image and the original galaxy do indeed have the same angular size.

If we, on the other hand, position ourselves so that the image is directly between us and the lens, then we can see it, even without a screen to project it on. Unlike the virtual image formed by the magnifying glass, this image is on the same side of the lens as we are, and we don't need to trace back light rays to a projected point; these light rays come directly from the image. This type of image is therefore called a real image. Provided we maintain the real image exactly between us and the lens, we can examine it just as we would any actual object. In particular, we can hold a magnifying glass to it.

This is just how a telescope—to be precise, a refracting telescope—works. The main lens, usually called an objective, creates a real image of the galaxy in a convenient place so that we can hold up a magnifying glass—usually called an eyepiece or ocular in the context of a telescope—and look at the image close up. The shorter the focal length of the eyepiece, the closer up we see the real image, and the larger the virtual image formed by the eyepiece. How much larger the image is than the actual galaxy (in terms of angular size) is given by the magnification of the telescope.

What's more, we can come up with a general rule for calculating the magnification produced by the telescope. Remember that the real image that Janus sees formed by the objective is the same one you look at through the eyepiece. At first glance, this situation might seem completely symmetrical.

Again, however, there is one important difference. Janus sees the real image from a distance equal to the focal length of the objective. You, on the other hand, see it from a distance equal to the focal length of the eyepiece.

Generally speaking, eyepieces have focal lengths that are much shorter than the objective. A typical objective has a focal length of perhaps 800 mm. An eyepiece might have one that is much shorter—say, 10 mm. In that case, the same image that Janus sees from 800 mm, you see from only 10 mm. And since you see it from a vantage point 80 times closer, it looks 80 times larger to you, too.

But—and here is the key—the real image has the same angular size as the galaxy to Janus's unaided eye. That means that the view of the galaxy you get through the eyepiece is 80 times larger than the one you get with the unaided eye. In other words, the telescope gives you a magnification of 80 power.

In all of this reasoning, it didn't matter precisely what the focal lengths of the objective and eyepiece were; all we really did to obtain the magnification was to divide the focal length of the objective by the focal length of the eyepiece. Even if we change those lengths, the magnification will continue to be the ratio of those two focal lengths. We can therefore get any magnification we like by increasing that ratio. (In particular, that means that department store telescopes can claim ridiculous levels of magnification just by making eyepieces of very short focal lengths, and—unfortunately—generally poor quality to boot.)

I should point out that the very first telescopes, invented at some point in the late 16th century in Holland and first put to astronomical use by Galileo, were not quite of this design. They used a different design that is similar but inferior in most ways. The first person to put together a telescope according to the principles described above was none other than Kepler.

The fact that we can treat the objective and eyepiece essentially separately—connected only through the real image formed by the objective—means that just about any eyepiece can be used with any telescope (even one of a different design), as long as they use the same barrel size.

This flexibility is not just a matter of convenience. You might think that what you'd want with any telescope is magnification and more, more, more of it, but there are plenty of reasons why that isn't always the case.

One is that the telescope's optical quality might not be up to it. Increasing the magnification will magnify the image, yes, but it will also magnify whatever flaws exist in the telescope as well. Even if the telescope's optical quality is irreproachable, the atmosphere might be too turbulent. Magnify the image too far, and it will look as though you are observing through a swimming pool.

But even if the telescope is of the highest quality, even if the atmosphere cooperates fully, different observing targets still demand different magnifications. The planets, for instance, are so small in terms of angular size that they will generally take all the power that you can throw at it. But other objects, such as the large star clusters, are too large for high powers. Their attraction lies in the fact that they are more concentrated than the generally sparser background stars. If you magnify them too far, all you see is the central condensation, and you lose the contrast that makes the clusters beautiful.

The upshot is that you generally want a range of magnifications, and you want to more or less evenly cover that range with your eyepieces. This leads to a common question amongst beginning amateurs: How do I select which focal lengths I should get for my eyepieces?

I'll take an unusual look at that question in my next essay.

Copyright (c) 2004 Brian Tung