by Robert A. Preston

<rbrtprstnearthlink.net>

Direct to: Seeing, Contrast, Stellar Diffraction, and M57

(an interpretation of ideas discussed by Roger N. Clark in the book, "Visual Astronomy Deep Sky", Cambridge University Press (England) and Sky Publishing Corp., Cambridge, Mass. (USA), 1990. This article may be freely distributed not-for-profit media, with proper attribution. Copyright, Robert A. Preston, 1998)

After re-reading Dr. Clark's 2nd chapter, "The Human Eye", (also chapters 3, 4, 6 and appendices E and F) more times than I care to admit, the basic premises underlying his ideas about visual detectability finally congealed in my mind. It's not that I'm dull-witted. It's just that Dr. Clark's subject is quite tricky and his writing style is decidedly unfriendly (at least it is for this particular reader). Now that I finally have a decent handle on his ideas, I'd like to share my thoughts with other astronomy buffs. Why? Because I think that Dr. Clark's ideas about visual sense perception belong in any visual astronomer's toolkit of essential facts. Dr. Clark's book is one of the most important books an amateur astronomer could read, in my opinion. Also, there are still a few kinks in my interpretation, so it would be useful to have some public debate about these issues.

I'll try to explain what I think is the most important idea in Dr. Clark's book, a concept that Clark calls the "optimum magnified visual angle". Clark thought that no one had ever described that concept in print. I suspect that's probably true, and I am herewith renaming the "optimum magnified visual angle" the "Clark angle" in honor of his description of it.

The Clark angle is (roughly speaking) the angular size to which a small object must be magnified in order for the object to have the highest possible visibility to a human eye. "Small object" can mean either a whole object, like a faint, fuzzy, elliptical galaxy, OR any PART of an object, for example, a portion of one of the spiral arms of galaxy. That is, all whole objects can be considered to be quilts composed of smaller "partial objects" of arbitrary size. This quilt-of-smaller-pieces concept is an important one, because it means that higher magnifications are usually needed to get the best visibility of parts of an extended object as opposed to the best visibility of the object as a whole. That is, the Clark angle of maximal visibility can NOT be obtained simultaneously for both a whole object and its smaller parts in one eyepiece. Maximal visibility of the smaller parts would require higher magnification than maximal visibility of the whole, and that would require changing to an eyepiece that gives higher magnification. This is a crucial point for visual observation of the structural features of galaxies and nebulae.

Yes, there is indeed an optimum magnification for making a small object as "visible" as possible. No, low power is NOT usually best for detecting detail in deep-sky objects, or for detecting the objects at all, contrary to the opinions of many amateur astronomers. I certainly thought that these were very strange ideas when I first read about them in Clark's book. After all, it was pretty obvious to me that faint galaxies are brighter at lower magnifications, and that makes sense, because their brightness is concentrated in a smaller area at the lower mags. Obviously if they are brighter at lower mags, they must be easier to see at lower mags, not higher mags as Clark was claiming. Clark's claims seemed pretty absurd, at first sight. But a careful study of his analysis of visibility factors convinced me that he's quite right. Anyone can prove this for himself or herself by simply looking carefully at several deep-sky objects, examining each with a variety of magnifications. That's a far easier method of proof than trying to understand the extremely convoluted logic of Clark's analysis. I have remnants of a terminal faith in logical analysis, however, so I tried to find out whether or not Clark's math was right. It is right, as far as I can tell, and the crux of the whole issue lies in the first derivatives of the various curves shown in Clark's figure 2.6.

Before I try to do with words what you could better do by actual experiment, I'll suggest a reason why so many amateur astronomers mistakenly believe that telescopes with low f-ratios ("rich-field telescopes") and/or low powers are the best tools for viewing deep-sky objects. That way, if you don't believe my words and you're too tired to look for yourself and you still cling to wrong ideas about magnifications, then at least you'll be able to understand why your wrong ideas might have formed. If you're already more than willing to learn about Dr. Clark's counter- intuitive ideas, you can skip the next paragraph. More importantly, if you're already willing to go out and look for yourself, with your telescope, then you could easily skip the rest of this article and spend the time looking at actual deep-sky objects. If you have the same problem with logic that I have, you'll read further.

The main reason for misplaced faith in the power of low powers was already described above, in my own initial reaction to Dr. Clark's book. That is, for reasonable people, even for those with far- above-average observational skills (like me!), it is intuitively obvious (and easily seen) that objects get dimmer as the magnification goes up. The fallacy in that view (so to speak) is that people's impressions about visibility are formed very early in their careers, at that particular time when they're looking at the very brightest showpieces in the night sky. The brightest objects are the ones that a novice can find in less than an hour of painful searching, and they're the only ones that can be easily seen even at magnifications as high as most people's eyepiece collection can provide. They might get pretty darn dim at the higher magnifications, but, hey, that's just "the price ya pay" for high magnification. It's a perfectly rational extrapolation of that novice's experience to deduce that high magnification should ALWAYS be avoided in any attempt to detect dim objects or see them at their best. It's a rational extrapolation, but it's an extrapolation that's based on a novice's impression of the bright core regions of a few easily found objects. And, as I will explain below, it simply doesn't apply to the visual detectability of any object that lacks bright regions, or to the visibility of details in small areas of an object, like the spiral arms of galaxies. A novice observer responds to overall brightness, not to the visibility of dim details. Then, using a faulty (but quite rational) extrapolation from first experiences, many observers never bother to spend the time to look for details that are visible only at high magnifications and in relatively dim views. These observers already "know" that raising the mag is the wrong way to go, and it's almost impossible for them to break free from this beginner's trap. At least that's the way it was for me, until I discovered Roger Clark's book.

Oddly enough, a key to an intuitive understanding of the Clark angle comes from considering the visibility of stars (other than the Sun!) in broad daylight. Most people (including most professional and amateur astronomers) have no idea that many stars can be seen in a telescope even at noon. Using a well-aligned and pre-focused 8-inch LX200 in a reasonably clear sky at noon, it is easy to GOTO and see stars as dim as 2nd magnitude, even in the eastern U.S., where really clear skies are very rare. I don't know how dim a star one could see with an 8-inch LX200 in a very deep blue, clear sky. It wouldn't surprise me if mag 6 or 7 or even 8 were detectable in an 8 to 12-inch telescope in very clear skies (assuming precautions were taken to avoid sun-induced tube currents in the OTA and other sources of thermal distortion that are common during the daytime, and last but most important, assuming that high magnification were used). (see endnote 1)

That point (above) about using high magnification in daytime introduces the first of two facts that are essential for understanding the Clark angle. Namely, magnification results in a darkening of the background sky. (It also darkens the object of interest, except for point-source stars that (effectively) remain equally bright at any feasible magnification - but lets not get ahead of ourselves). Daytime skies, or light-polluted nighttime city skies, or even dark country skies - they ALL get darker, necessarily, as the mag. goes up. That fact is simply a necessity of ordinary physics. Thus, the stars will always come out at noon as soon as you use enough magnification to darken the background sky enough. (see endnote 2)

The second fact that's required for an intuitive understanding of the Clark angle is this: the ability of the human eye to detect a smallish object (viewed against a neutral backdrop) increases by about a thousand-fold as the angular size of the object increases. By "smallish object", I mean an object with an angular diameter between 1/2 arcminute and about 6 degrees. To get a sense of the size of the "smallish" objects I'm talking about, consider this. I'm writing and reading this article with the text 50 cm away from my eye. At that distance, 1/2 arcminute means an object about one-tenth millimeter across, while 6 degrees means an object 5 centimeters across. The detectability of two objects that have those two angular dimensions differs by a factor of about 1000, according to measurements done by H.R. Blackwell in 1946 (endnote 3). (Blackwell reported only modest improvement in detectability as an object increased in size beyond 6 degrees). Blackwell's observations have such important consequences for visual astronomy (and for target shooting at night), that I'm going to call the size-dependence of visual sensitivity "the Blackwell effect". Please note that the angular sizes that are involved in the Blackwell effect are quite small. At most, around 6 degrees, the size is only about one-eighth of the width of a typical 50-degree eyepiece field. At the small end, 1/2 arcminute, the size is near the limit of visual detectability even for rather high-contrast objects. Thus, the Blackwell effect includes the entire range of sizes we would normally think of as "fine detail".

Armed with the previous two paragraphs, I am now poised to explain the existence of the Clark angle, as soon as I set the observer's stage. Imagine that you are looking at a very small, dim galaxy in your telescope (not really a microscopic galaxy - one that just LOOKS small because of its distance.) (It's important to me that I don't lose sight of (!) the fact that ANY galaxy is larger than I can imagine and has millions or billions stars plus an unknown number of black holes plus some things that humans have yet to name and know). Also imagine that this galaxy is one of those deep sky objects that you can just BARELY see, even with averted vision and eyes that have been dark-adapted for an hour under country skies so utterly dark that you can't see your hand in front of your face (in short, a "Preston object", to coin yet another eponym). Finally, suppose that you are using an eyepiece that provides 80x magnification and that the true angular diameter of this galaxy is one arcmin, so it appears to be 80 arcmin diameter in the eyepiece field (in retinal size, equivalent to that of a one-cm line at 50-cm distance).

The stage is set. Now, what will happen to the visibility of this Preston object when you raise the magnification three-fold, to 240x? The brightness of BOTH the galaxy and the background sky will decrease by an amount (240/80) squared, or 9-fold. This simultaneous decrease in brightness of the galaxy AND the background sky will leave the physical contrast between the two utterly unchanged. IF physical contrast were the only factor to consider, the galaxy would be every bit as difficult to see as it was at the lower power. BUT we still need to factor in the Blackwell effect. Because the galaxy is now 240 arcmin diameter instead of 80 arcmin, our human eye will find that it's about two or three times easier to see (based on Blackwell's measurements as reported in Clark's book, chapter 2, figure 2.5 and 2.6). Viola! The galaxy is no longer teetering on the edge of visibility. Instead, it looks quite obvious and certainly real, beyond any doubt. Thus, we've increased our ability to see a dim object simply by using a high enough magnification.

It's possible to have too much of a good thing. Magnification beyond the Clark angle is like that. Since the retina begins to get bored (or something) when an image approaches and then exceeds 6 degrees in width (according to Blackwell's measurements), magnifying the image much larger than 6 degrees can't increase its visibility. In fact, since magnification always makes things dimmer as well as larger, excessive magnification merely takes the total picture (the object PLUS its background) closer to the eye's light detection threshold. According to Blackwell's empirical results, as discussed by Clark, excessive magnification (i.e., magnification larger than the Clark angle) decreases the detectability of a low-contrast object against its background.

After recovering from the notion that things have to be magnified a definite optimal amount for the best detectability, I was again sent reeling by Clark's contention that the optimum magnification for detecting an object is INVERSELY related to the aperture of the telescope used to view the object. Yup. The larger the aperture, the LESS magnification is necessary to obtain the optimal visibility of any dim object. This result is implied by Clark's graphs and equations, but it is possible to grasp the matter intuitively, too. Clark said that "the larger a telescope is, the greater an object's surface brightness is at any given magnification, so it can be detected at a smaller apparent size." That explanation, by itself, did not make sense to me, since it seems contradictory to the previous discussion. However, Clark's explanation did make sense when I realized that it applies specifically to very dim objects, not just any old object. In other words, he's talking only about objects dim enough to be affected by the Blackwell effect. R. A. Greiner (see endnote 5) explained that, for objects that are dim enough to be subject to the Blackwell effect, "there is a struggle between the eyes' ability to see contrast, which gets poorer with decreasing brightness, and the eyes' improving ability to see contrast as the image gets larger". It is the balance point in that struggle that changes with a telescope's aperture, so that less magnification is required for OPTIMUM visibility of the brighter images that are provided by larger apertures.

The practical consequence of the aperture/optimal magnification relationship is that the size of the Clark angle for any given deep sky object depends on the aperture used to look at the object. Thus, it's wrong to think that, for optimal visibility, any object should be magnified until it is 6 degrees in size. The optimal size depends on the aperture used, among other things. In fact it's almost impossible to calculate the optimal magnification for any deep sky object, because the calculation depends on the brightness distribution across the object (not often known) and the size of the part of the object that is of most interest to the observer. In reality, a whole range of magnifications are required to best see an object and the smaller details within the object. The value of the concept of the Clark angle is not so much for predicting an optimal magnification, but for suggesting that a wide range of magnifications is useful, including very high magnifications in some cases.

Neither Clark nor Blackwell provided a biophysical explanation for the Clark angle-- it's just an empirical observation (as far as I know - I have not had time to read the original Blackwell article, yet). I suspect that the Blackwell effect reflects the shape of the spatial distribution of rod receptors on the retina. At the low end of Blackwell's range of sizes (0.5 arcmin), the retinal image illuminates only one or two rod receptor cells at best. At the high end of the size-range (6 degrees) the retinal image could provide (as near as I can estimate) the greatest possible spatial density of illuminated rods. Physiological experiments might address this question one day. (endnote 4)

As far as I know (and, admittedly, it's not very far), very few people understand the visual implications of the Blackwell effect and the Clark angle. If the above discussion of the results of Clark and Blackwell stimulates someone to explain the biology of the Blackwell effect and the Clark angle, this article will have served a minor purpose. If it stimulates amateur astronomers to explore the use of high magnifications to see more of the detail present in the night (and daytime!) sky, then that would be just peachy, indeed.

Endnote 1

The visual detection of low contrast objects during the day follows rules somewhat different than those that apply at night. The differences result from the fact that entirely different populations of retinal receptor cells operate during the day and night (cones by day, rods by night). The two different receptor types have different sensitivities to light as well as vastly different distributions on the retina. The bottom line is that "averted vision", so useful for detecting faint things at night, is worse than useless for detecting small objects during the day (if you have any doubts about this, look at the middle of this line of type and try to figure out what letter of the alphabet is at the front of the line). During daytime conditions, one needs to gaze almost precisely AT a small, low contrast object in order to see it at all - the useful daytime visual field for seeing stars is only one or two degrees wide. And this is where I get to be "on-topic" for the MAPUG email list. A carefully aligned LX200 fitted with a high-power eyepiece that is prefocussed using the moon or the sun (use a solar filter during focusing on the sun, please, unless you want to broil your cornea for the sake of seeing an insignificant star in the middle of the day--a feat that almost certainly would qualify as the least useful act any human ever performed) can GOTO a particular star with an accuracy that means that only a small amount of visual scanning of the center of the field of view is necessary in order to detect the star. Without an LX200 (or a very accurate set of digital setting- circles), this is an essentially impossible task for any object less bright than Venus or maybe Sirius.

Endnote 2

At some point, as the magnification goes up and the actual sky background brightness goes down, the actual brightness of the sky will fall below a level of "noise" brightness in the eyepiece that is caused by stray light. Stray light is always present in the light path because of the limitations of telescope designs. Bad designs let so much stray light into the eyepiece that it is impossible to make the background truly dark by magnifying the sky. In other words, magnification of the sky itself can never make the apparent background in the eyepiece any darker than the level of stray light, whatever level that may be. The reason telescopes are painted black inside (and cleverly designed baffles are installed in some) is to reduce the amount of stray light as much as possible so that high magnifications (of the sky) will indeed darken the apparent background in the eyepiece and make higher contrast possible.

Endnote 3

Blackwell, H. R., 1946. Contrast Thresholds of the Human Eye. J. Opt. Soc. Amer. 36:624-643.

Endnote 4

Readers interested in rod and cone distributions can find much useful information in Clark's 2nd chapter.

Endnote 5

Doc G is an amateur astronomer in Madison, Wisconsin. He discusses astronomical aspects of visual perception on a page at his website.

Seeing, Contrast, Stellar Diffraction, and M57

Robert A. Preston
July, 1998, Pittsburgh PA

Recently on the e-mail list for the Meade Advanced Products Users Group (MAPUG), we discussed visual detection of the central star in the Ring Nebula M57 (NGC6720) in the constellation Lyra. Factors that affect the visual detectability of the central star were thought to include telescope aperture, magnification, seeing conditions, sky transparency, and the experience level of the observer. Opinions differed about the relative importance of each of these factors. As a result of the discussion on MAPUG, I'm convinced that I really don't know the facts of the matter. Be that as it may, I have some new ideas about this that I'll throw out for any idea-feeding fish in the area.

In "Visual Astronomy of the Deep Sky", Roger Clark makes this interesting comment about the central star in M57: "it is suspected of being variable because it is sometimes easy in a 12-inch telescope, at other times difficult in a 40-inch." I guess it might be a variable, as Clark suggests - do supernova remnants typically become variable, however? I wonder if variability of its visibility has a different explanation. I wonder if it is merely a seeing- and magnification- limited object, visualized best under a crucial combination of not-so-common conditions.

Roger Clark trashed the commonly-held notion that 50-60x per inch of aperture is the maximum useful mag. for any given telescope. That limit is true ONLY for bright images. Based on a measured resolution value of about one-half degree for dim objects (at the threshold of visual detectability), Clark calculated that 330x per inch of aperture is required to maximize the resolution of very dim objects by the human eye. Thus, depending on the brightness of the object, anywhere from 50-330x per inch of aperture is required for maximum visual resolution.

Experienced observers will immediately notice that average seeing conditions make even 50x per inch of aperture pretty useless most of the time (except, perhaps, with 2- or 3-inch apertures). What's more, even if seeing were perfect, very high mags bring out the worst in telescope mountings. Trying to use a resolution-optimizing dim-object mag. of 1000x with a three-inch refractor would quickly reveal whether minor imperfections in the drive mechanism or tripod (or the local geology-freight trains within a mile or two can matter) have become the resolution-limiting factor.

What about imperfect seeing? What kind of imperfection are we talking about? There's "fast seeing" and "slow seeing" and kinds of seeing that haven't even been named yet. These do different things to the diffraction pattern of a star in a telescope. Depending on exactly what the ambient seeing conditions do to the shape of the light intensity distribution in the Airy disk in the center of the diffraction pattern of the M57 star, I guess that the surrounding nebulosity might or might not "erase" the star's visual detectability in any given telescope, whatever the magnification used. All that would be required is for the peak brightness at the center of the Airy disk to be smeared around enough to make the peak surface brightness of the Airy disk about equal to the surface brightness of the surrounding nebulosity. Is there in fact a type of "seeing" that would do that? I don't know, really, but I suspect it would be quite common.

The central star in M57 is mag 14.8, according to Roger Clark. The ideal limiting magnitude for an 8-inch telescope is 15.2 (assuming the scope has a transmission factor of 0.70 and a negligible stray light level (the latter not true, most likely, since my scope hasn't been flocked). This means that it might be possible for me to see the central star in my 8-inch scope. I could run the mag up as high as 8x330, or 2640x, in an attempt to darken both the central star AND the background nebulosity in the "hole" of M57 to the required threshold level of detectability. An eyepiece with a focal length of 5 mm gives only 400x in my f10 scope. (If it were an f15 Maksutov, I'd have 600x!) OK, darn it, I guess I'll get one of those 3x Barlow-thingies. Then I'll have 1200x. That might do it, if it reduces the brightness of the central star and its associated nebulosity enough. I might have to disconnect the drive to kill its vibrations, I guess. I might have to put my heart-beat on hold for a while to kill those vibrations, and I might have to wait a while for that rare patch of perfect seeing to float by. It can't be any worse than hand-guiding an astrophoto for hours on end. But, you know, I think I'll flock the scope before spending too many hours staring at M57 at 1200x.

Maybe I should just upgrade to the 10" LX200, which would have an "ideal limiting magnitude" (flocked) of 15.7, according to Clark....Nah, no way. I'd have to be a bit nuts to trade the portability of my 8-inch for a 10-inch albatross just for the pleasure of sitting and staring for hours trying to momentarily glimpse the star in M57 during an instant of good seeing.