Figure 1. A Barlow lens intercepts the converging light rays and
magnifies the real image created by the telescope objective. (Not to
scale.)
May not music be described as the mathematics of the sense,
mathematics as music of the reason?
The musician feels mathematics, the mathematician thinks music:
music the dream, mathematics the working life.
--James Sylvester (1814-1897)
SOME MEASURE of my esteem for the French amateur mathematician and physicist Pierre de Fermat (1601-1665) is the great frequency with which he shows up in my essays. (No, I'm not going to list each of his appearances. You'll have to look for them yourself.) I suppose he is best known for his Last Theorem, but he has also shown up for his Principle of Least Time, both of which appear in at least one of these essays.
One of his lesser-known results which I'm pretty sure I haven't mentioned is the following. Take any prime number p, and any number n which is not a multiple of p. Fermat asserted, without proof (which was his wont), that the remainder when n p is divided by p is equal to the remainder when n is divided by p. In mathematical notation, we say
where n mod p (read "n modulo p") means the remainder when n is divided by p. The great Swiss mathematician Leonhard Euler (1707-1783), who begged off an attempt to prove Fermat's Last Theorem, did succeed in proving this result, which has ever since been called Fermat's Little Theorem. (Euler was the first to publish this result. Apparently, Leibniz, the co-inventor of calculus, also proved it, but left it unpublished in one of his notebooks.)
Of what use is such a result? Why would Euler bend his considerable intellect to something that looks like little more than numerical tinkering? For over two centuries, anyone looking in would have judged that Euler had basically wasted his time. Newton had invented the calculus to solve a fairly theoretical problem, too, but it had found great use in engineering and physics. To what could Fermat's Little Theorem be applied?
Yet, as the old saying goes, all things come to him who knows how to wait. In the last third of the 20th century, important encryption schemes were developed that depended on certain properties of prime numbers, and fast computation of those properties. It turned out that Fermat's Little Theorem had direct applicability in this area. So Euler didn't waste his time after all. (Interestingly, the proof of Fermat's Last Theorem relies on certain properties of what are called elliptic curves, which also can be employed to speed up some encryption schemes. There's no other connection between the two, though.)
I mention all this because, naturally enough, I have a little idea I would like to share that, as far as I know, has no conceivable applicability in real life. I'm not so arrogant as to think that it will one day somehow become important somewhere--but one can always hope.
As I mentioned in my last essay, the magnification that you get out of an eyepiece and a telescope is the focal length of the telescope's objective divided by the focal length of the eyepiece. I also mentioned that as much as you might think that high magnification is the reason for being of a telescope, there are plenty of objects for which lower powers are desirable.
One reason for this is that some objects are simply too big to fit into the eyepiece field of view at high power. Consider the Moon, for instance. Its angular size is one-half degree, meaning that if you were to stack 180 Moons on end, they would span the right angle from the horizon to the zenith, since there are 90 degrees in a right angle.
If you use an 8 mm focal length eyepiece in a telescope whose objective has a focal length of 800 mm, you'll get a magnification of 800, divided by 8, or 100 power (usually written 100x). That means that the magnified Moon will have an apparent size of 100 times half a degree, or 50 degrees. It so happens that many eyepieces have an apparent field of view (often abbreviated AFOV) of about 50 degrees. In that case, the 50-degree magnified Moon will just barely fit into the field of view.
You can get higher magnification on the Moon if you use an eyepiece of 4 mm focal length. That yields 200x, magnifying the Moon to an apparent size of 100 degrees. However, unless the eyepiece has a distinctly different design, it will likely have an apparent field of view of 50 degrees, just like its longer brother. Now the magnified Moon is too big to fit into the field of view, by a factor of two.
In the case of the Moon, there are plenty of small details to look at, so the fact that the Moon doesn't fit into the field isn't a big liability.
That isn't necessarily the case, though, with other objects where a large part of the attraction is the way that they sit against the background of stars. For instance, there are congregations of stars, called clusters, where there are more stars in a given volume of space than in our galaxy generally. Ideally, that's just the way they'll look in the eyepiece: a central concentration of stars, surrounded by relative dearth.
So long as the cluster's magnified size is smaller than, say, 15 or 20 degrees, the cluster is set off nicely from the background stars and it is reasonably evident where the cluster begins and ends. But if you magnify it so much that it extends beyond the eyepiece's AFOV, all you see is the cluster's concentration of stars, and the contrast is mostly lost. In other words, it isn't the cluster itself you need to see--it's the contrast between it and the rest of space.
For this and other reasons, it's useful to have a range of available magnifications from which to pick and choose, depending on what it is you're observing (as well as the sky conditions, but I'll save that for a later essay).
One way to get such a range is to buy a so-called zoom eyepiece. These eyepieces are generally made up of two groups of lenses that can be moved closer together or further apart. If you separate them, they behave the same as one eyepiece with a short focal length; if you bring them closer together, they behave as though they were a long focal length eyepiece. Since the distance between the two groups can be varied smoothly (usually by means of a helical screw design), the effective focal length of the zoom can also vary smoothly.
The zoom sounds like the ideal solution, but there's a catch. In order to form an accurate and pleasing image, the shapes of the lenses in eyepieces often has to be controlled quite tightly, in a way that depends on the spacing between them. But in a zoom, where the spacing changes, there will be some effective focal lengths for which the lens shapes work well, and others for which they don't. It's often thought, therefore, that zoom eyepieces are really a compromise solution, giving smooth variability in magnification at the expense of exquisite image quality.
To be sure, recent years have brought out a number of complicated zoom designs that deliver performance approaching that of fixed focal length eyepieces, but there's a catch there, too: they tend to be rather expensive.
The alternative, then, is to get a collection of eyepieces of differing focal lengths. Each one will only yield a single magnification in any given telescope, but if you have enough of them, you can cover a range of magnifications well enough. After all, there isn't likely to be a significant difference between a view at 80x and one at 85x. That raises the question: Which eyepieces should I get? And since for any telescope, each eyepiece determines a single magnification, the question is really, what set of magnifications should I have?
One property the magnifications ought to have, ostensibly, is that the range of magnifications should be covered evenly. That is, there shouldn't be any large gaps; conversely, there shouldn't be two or more eyepieces yielding very similar powers.
That still leaves some room for judgment. For instance, consider the following two series of magnifications:
20x, 40x, 60x, 80x, 100x, 120x, 140x, 160x, 180x, 200x, 220x, 240x
20x, 25x, 32x, 40x, 50x, 65x, 80x, 100x, 125x, 160x, 200x, 250x
Both series are evenly spaced out, but each in their own different way. In the first series, each magnification is 20 higher than the previous one. Such a series, in which the numbers are separated by a constant difference, is called an arithmetic series.
In the second series, each magnification is approximately five-fourths times as much as the previous one. (There are small rounding errors, which for our purposes I'm going to ignore.) This kind of series, in which successive numbers are related by a constant ratio, rather than by a constant difference, is called a geometric series.
At first glance, there doesn't seem to be much that would help us decide which series we should choose. Neither series has any noticeable gaps or unnecessary duplications.
We might, however, make one observation: An object 1.5 degrees across looks just as big at 20x as one 9 arcminutes does at 200x. If you do the multiplication, you'll see that both will appear magnified to an angular size of 1,800 arcminutes, or 30 degrees. In either case, you might want to increase the magnification to see some additional detail.
Suppose we chose the arithmetic series. From 20x, the next available magnification is 40x, or twice as much. But twice as much might be too far. It would, for instance, put the object's apparent size at 60 degrees, which might not fit in the eyepiece anymore. On the other hand, from 200x, the next available magnification is 220x, or only 10 percent greater. The new apparent size would just be 33 degrees; the view wouldn be changed hardly at all, and you would have to go perhaps one or two steps further before noticing much of a difference.
But suppose we chose the geometric series instead. Whether you're looking at the 1.5-degree object at 20x or the 9-arcminute object at 200x, the next available power is 25 percent higher, bumping the size of the object up to 37.5 degrees. Not so much that the object won't fit into the field of view, but not so little that there isn't any noticeable difference, and of course, the change in appearance is much the same for both objects. For this reason, it's my own opinion that using a geometric series makes more sense.
If we settle on using the powers in the geometric series, we can easily figure out which focal length eyepieces we should get. We simply divide all of the magnifications into the focal length of the objective, which we'll say is 800 mm, as before. In that case, we get the series
40 mm, 32 mm, 25 mm, 20 mm, 16 mm, 12.5 mm, 10 mm, 8 mm, 6.5 mm, 5 mm, 4 mm, 3.2 mm
where, again, some of the numbers have been rounded off for convenience, and also because--especially in the case of the longer focal length eyepieces--they match commonly available products. You'll notice that the focal lengths, like the magnifications, are related by a constant ratio. The only difference is that before, the constant ratio was five-fourths; now, it is four-fifths.
One thing that might seem unpalatable about this list of eyepieces is that there are so many of them: a dozen in all. Most of us don't have room for lugging around a dozen eyepieces; many of us don't have the money to buy them in the first place.
Such a thought might possibly have occurred to the English scientist Peter Barlow (1776-1862), when he collaborated with his optician friend George Dollond in creating an auxiliary lens for telescopes. More directly, Barlow was interested in a way of making it easier to measure the angular separation between the individual stars in a double star. In many cases, even the shortest focal length eyepiece available didn't yield a high enough power to distinguish the stars clearly. What Barlow wanted was a way to achieve a higher magnification without having to manufacture a new telescope or a new eyepiece.
With this in mind, he and Dollond designed and assembled a negative or concave lens. Such a lens makes parallel light rays diverge, just as a positive or convex lens (what the telescope objective and eyepiece are) makes parallel light rays converge.
However, Barlow and Dollond didn't put the lens in the path of any parallel light rays. Instead, they placed it inside the telescope, where it intercepted the converging light rays before they had yet formed an image. By diverging the light rays, it magnified the size of the real image. (See Figure 1.)
Figure 1. A Barlow lens intercepts the converging light rays and
magnifies the real image created by the telescope objective. (Not to
scale.)
As a result, the view of the object as seen in the telescope was magnified compared to the view without the added lens. Barlow and Dollond could increase the magnification without making a new telescope or a new eyepiece. Admittedly, they did have to manufacture a new lens. However, this lens--now called a Barlow lens--can be used with any telescope and any eyepiece to increase magnification in all combinations. In effect, it can double the size of any eyepiece collection.
The degree to which magnification is increased or amplified depends on how close the Barlow lens is placed to the focal point. If it is placed close to the focal point, the divergence effect doesn't have enough space to magnify the size of the real image significantly, so the amplification is relatively small. If, on the other hand, you put the Barlow far in front of the focal point, there's enough space for the lens to magnify the real image considerably, and the amplification is relatively large. (If this sounds eerily similar to the way that zoom eyepieces work, that's because that is how some zoom eyepieces work.) You can therefore get any degree of amplification you like by changing internal spacings, but the most common case today is to amplify the magnification by a factor of two; this is often called a 2x Barlow.
You can therefore think of any given Barlow-eyepiece combination has having a focal length just one-half of the eyepiece in isolation, because such an eyepiece would have double the magnification. For example, a 2x Barlow in conjunction with a 16 mm eyepiece is equivalent to an 8 mm eyepiece, because both yield 100x when used in our 800 mm focal length telescope.
In fact, we can, by judicious use of the 2x Barlow, cut the size of our eyepiece collection in half. We could, for instance, have eyepieces of the following focal lengths:
40 mm, 32 mm, 25 mm, 10 mm, 8 mm, 6.5 mm
which is only six eyepieces. If we use this collection in conjunction with the 2x Barlow, we get the following effective focal lengths:
20 mm, 16 mm, 12.5 mm, 5 mm, 4 mm, 3.2 mm
and the two sets of focal lengths together make up our original twelve. Somewhat less efficiently, we could choose the following eyepieces:
64 mm, 40 mm, 25 mm, 16 mm, 10 mm, 6.5 mm, 4 mm
which is only seven eyepieces, and the 2x Barlow will then effectively give us these:
32 mm, 20 mm, 12.5 mm, 8 mm, 5 mm, 3.2 mm, 2 mm
This uses one more eyepiece, but on the other hand, it gives us two more focal lengths (64 mm and 2 mm) to use at the telescope.
The reason this works as well as it does is an interesting coincidence that was known as far back as the ancient Greeks.
Music played a prominent role in Greek art, and Greek life in general. It undoubtedly had a seminal influence on many other musical forms that came up in and around the Middle East during the time of Christ and for centuries thereafter. We don't, unfortunately, know just how strong an influence it was because most ancient Greek music was lost forever--in particular, during the final burning of the Alexandrian Library around the year 400.
We do know that the Greeks saw an unbreakable link between music and mathematics. It was Pythagoras (c. 569-475 B.C.) who first recorded this relationship in a concrete way. He was experimenting with the way that the length of the strings in a string instrument, such as a lute, affected the pitch of the notes the strings sounded--that is, how high or low the notes were.
Pythagoras found that when the string was effectively cut in half (by pressing it against a fingerboard), the pitch of the new note harmonized well with the original; it was, in fact, a "copy" of the original, but sounded one entire scale above. Think of the two do's in the scale, do re mi fa sol la ti do. Because these two do's span, inclusively, eight notes, the distance or interval between the notes is called an octave, after the Latin word for "eight."
Pythagoras also experimented with other string lengths, and he found that if he took a string that sounded the note do, and cut it so that it was two-thirds as long, the new string would sound the note sol. If he cut it so that it was three-fourths as long, it would sound the note fa. And if he cut it so that it was four-fifths as long, it would sound the note mi.
If you look at these fractions, you'll note that they are internally consistent (as they must be). For instance, in musical terms, we say that the intervals do-sol and fa-do are the same, in the sense that sol is just as much higher than the low do as the high do is higher than fa. That means that the high do string should be two-thirds as long as the fa string, just as the sol string is two-thirds as long as the low do string.
In other words, if we start out with a string 60 cm long, playing a low do, then the fa string must be 45 cm long, and the sol string 40 cm long. And sure enough, the high do string, which must be 30 cm long, is in fact two-thirds as long as the fa string.
The do re mi scale, which is a product of medieval music, is not an evenly spaced scale, in the sense that not all pairs of adjacent notes form equal intervals. Two of the intervals, mi-fa and ti-do, are about half as wide as the others. (On a piano, these are pairs of white keys that do not have black keys between them; the other pairs of white keys all have one black key in between.) There is room for two of those intervals in between the notes do and re, for example.
Following a later convention, we can insert extra notes to make the scale evenly spaced, as follows:
do di re ri mi fa fi sol si la li ti do
where all the extra notes correspond to black keys on the piano. Such a scale is called chromatic, in contrast to the original scale, which is called diatonic. (There are reasons underlying each of those names, but if I get into them, I'll never finish.) Using the chromatic scale, we can see that the intervals do-mi, mi-si, and si-do are all the same. They must therefore represent three consecutive reductions of the string to a factor of four-fifths. But the entire interval do-do, we already saw, is a factor of one-half.
That means that if we cut the string to four-fifths the previous length three times, that should be equivalent to cutting it to one-half the length just once. In other words, four-fifths multiplied by itself three times should equal one-half. That isn't exactly true:
whereas 1/2 = 64/128, but that's close enough to make the musical intervals in question almost the same. To me, the fact that these simple fractions work out almost perfectly is a fairly surprising coincidence. I suspect, though, that Pythagoras and his students had such faith in the power of numbers that they found it a natural course of affairs.
Incidentally, the series of string lengths represented by the notes of the chromatic scale is a geometric series. Each successive cutting--from do to di, from di to re, and so on--produces a string related to the previous one by a constant ratio. If it were an arithmetic series, each string would be related to the previous one by a constant difference, and it would take only about 20 cuttings or so to reduce the original string to nothing.
Since there are twelve intervals in one full octave of the chromatic scale, that constant ratio is repeated twelve times. If we denote that ratio by the letter r, then in mathematical terms, r multiplied by itself twelve times equals one-half. That is,
Another way of saying the same thing is that r is the twelfth root of 1/2. Such an equation can be solved for the variable r, albeit not exactly: the best we can do is give an approximate decimal: r = 0.94387.... The musical interval corresponding to this ratio (such as do-di) is called a half step, since it takes two of them to make up the full step between do and re.
Using that terminology, we say that the do-sol interval is equal to seven half steps; the do-fa interval is equal to five half steps; and the do-mi interval is equal to four half steps. The musical coincidences that Pythagoras chanced upon can therefore be expressed, mathematically, as the fact that r7, r5, and r4 are approximately equal, respectively, to two-thirds, three-fourths, and four-fifths.
What all this has to do with our eyepiece series is that adjacent eyepieces have focal lengths that are related by a ratio of four-fifths. That is the same ratio involved in the musical interval do-mi, which is called a major third (because it spans three notes, do re mi). As we saw, three intervals of that size will fit almost exactly in one octave, which is a factor of one-half. That's why a Barlow lens fits so well into the series: used in conjunction with an eyepiece at one point in the series, it provides, virtually, another eyepiece that is three places later in the series.
We can recast our original eyepiece selections in terms of notes and musical intervals, as follows:
Figure 2. An eyepiece table comparing musical notes with eyepieces.
The Barlow lens "plays" the note one octave higher.
(Eyepiece focal lengths without Barlow shown in red.)
Each eyepiece corresponds to a note; for example, a 40 mm eyepiece corresponds to the note do1. In order to identify which do is which, we break up the musical sequence into octaves, and we use the superscript 1 to indicate that this particular do belongs to the first octave. In conjunction with a Barlow lens, that same eyepiece "plays" the identical note one octave higher. That is, a 40 mm eyepiece with the Barlow is equivalent to a single 20 mm eyepiece, which corresponds to the note do2--the do in the second octave.
All the focal lengths--including the native focal lengths of the eyepieces by themselves, as well as their effective focal lengths when used with the Barlow--are separated by multiples of a major third. Since three major thirds make up an octave, the use of the Barlow lens creates pairs of focal lengths, separated by three intervals on the focal length sequence.
We can do even more. There is a device called a focal reducer, which is similar to a Barlow lens, but which works the other way: it reduces the effective focal length of the telescope (hence the name). Alternatively, we can view it as increasing the effective focal length of any eyepiece. Like Barlow lenses, the magnitude of the effect varies, but a common value is 5/8. That is, the effective focal length of the telescope is reduced to 5/8 of the original, or equivalently, any eyepiece used with a focal reducer has an effective focal length 8/5 times its original focal length. (In astronomical parlance, such a reducer is called an f/6.3 reducer, because it turns a telescope whose focal length is 10 times its width--an f/10 telescope--into one whose focal length is 5/8 of 10 (or 6.3) times its width.)
It so happens that 5/8 is almost exactly equal to two major thirds. There is a major third from do to mi, and also a major third from mi to si, so a string that sounds the note si is very nearly 5/8 as long as one that sounds the note do. Two major thirds make up a new interval called a minor sixth (so called because it is a half-step shorter than a major sixth, from do to la).
If we incorporate this into our musical diagram, we find that each eyepiece now plays three notes: its original note; a second note when used with the Barlow, one octave above; and a third note when used with the focal reducer, a minor sixth below.
For that matter, one can use an eyepiece with both the Barlow lens and the focal reducer. This makes the eyepiece play a fourth note, one that is adjusted upward by an octave, then downward by a minor sixth. Since an octave is three major thirds, and a minor sixth just two, this fourth note must just be the original note, raised by a major third.
We can put all this into our eyepiece selection. Since each eyepiece plays four notes, the same range which at first required a dozen eyepieces, from 40 mm down to 3.2 mm, we can now cover with just four: 40 mm, 16 mm, 6.5 mm, and 4 mm. This is illustrated in Figure 3:
Figure 3. As before, the Barlow plays an octave higher (up three
places).
In addition, the reducer plays down a minor sixth (down two places);
both accessories together play up a major third (up one place).
We can't quite do it with three eyepieces, even though three times four equals twelve, because although the brackets fit together perfectly, our range of eyepieces isn't infinite. The brackets therefore don't all squeeze into our range, and we also need one extra eyepiece (the 4 mm one) to complete the coverage. Still, not bad for a little musical interlude!
Alas, I must admit that this pretty picture is mostly academic. One problem is that using Barlow lenses and focal reducers adds complexity to the process of switching focal lengths. If you only use eyepieces, then all that is required is to take one eyepiece out of the telescope, and insert another.
But under the scheme we ended up with, in order to switch from, say, 12.5 mm (16 mm eyepiece with Barlow lens and focal reducer) to 20 mm (40 mm eyepiece with Barlow lens alone), we have to take the 16 mm eyepiece out of the Barlow, take the Barlow out of the focal reducer, uninstall the focal reducer, reinsert the Barlow, and put the 40 mm eyepiece into the Barlow. Whew! This example is worse than most, but it highlights the potential problems. Many people would rather either pay more for the ability to use a few more eyepieces in lieu of the Barlow and reducer, or to put up with a less densely filled in sequence of focal lengths.
Another potential problem is that the eyepiece, Barlow lens, and focal reducer may not play well together; they may play "out of tune," so to speak, distorting the image in one way or the other. Since a main goal of fine-tuning the magnification is to detect details that wouldn't be visible at even similar magnifications, compromising the image to save weight in the eyepiece case is probably counterproductive.
One last problem is that the advantage of longer focal length eyepieces--a wider field of view--may be limited by the width of the eyepiece barrel. In the popular 1-1/4-inch format, no advantage in field of view is gained in using most eyepieces longer than about 35 mm in focal length.
It's too bad. I'm rather pleased by the way that all the focal lengths dovetail into one another, and I still hold out hope--however faint!--that someone will one day read this and make some unanticipated use of it. (But I'm not holding my breath.)
Copyright (c) 2004 Brian Tung