Another interpretation of the data from Blackwell, H R (1946): Contrast Thresholds of the Human Eye.

First a few words about why I write this. In 1991, I bought a 6in. reflector and started a "career" in amateur astronomy, still going on. Later the same year I found a very fascinating book, Visual Astronomy of the Deep Sky by Roger N Clark (Cambridge University Press and Sky Publishing Corporation, 1990). I found it extremely interesting, as it was the first I have read that had an in-depth discussion of the physiology of vision, as applied to amateur observing, and it made a very strong case for the use of quite high powers on deep sky objects. My limited observing experience supported this.

What's it all about - an introduction for those who haven't (yet) read Clark's book

Quite often (but hopefully less often today) it is claimed that faint nebulae or galaxies are best seen at low powers, because then they are "brighter" than at higher powers. Technically this is true, of course, and is valid for film exposures. The eye, however, has more complex ways of operation:

It has two rather distinct modes, "direct" and "averted" vision. Direct vision uses the central part of the retina, with about 1 degree of field, where detail is seen at best resolution. This is what you normally use to see detail when the background is not very dark. When it is, you may see fainter light if you direct your look a little to the side of the object (around 10-15 degrees). This is averted vision, and is regularly used to advantage by amateur astronomers.

It also has a very efficient built-in "gain control", known as dark adaptation. The sensitivity of the eye increases slowly if you only let it see very faint light, and it can take half an hour or even longer to approach maximum sensitivity for averted vision.

If you have a telescope and increase its magnification, this spreads out the light both from the sky background and the object, to make them seem less "bright". However, the contrast between object and background remains the same, and the dark adaptation will actually increase the sensitivity of the eye to let you detect fainter objects. If the object is magnified too much, the light will be spread out over a too large area within the eye to be "collected" efficiently, and sensitivity is lost.

Clark devises some very complex calculations to find the telescope magnification that lets you best see an object, given the telescope aperture, sky background brightness, and object size and contrast against background. Note that even the darkest country sky is quite bright, but light pollution from city lights makes it even brighter.

About magnitude

The magnitude concept has its advantages and drawbacks, but it is established. Pogson found that a logarithmic scale for brightness matched the ancient star magnitude scale reasonably well, if 5 magnitude steps corresponded to a brightness ratio of 100 to 1. Since magnitude 1 is brighter than magnitude 6, the magnitude difference is -2.5 times the decimal logarithm of the brightness ratio (the negative sign is rather unfortunate).

For extended objects such as nebulae or sky background, light can be measured in magnitudes per square arcsec, a unit that I prefer to abbreviate as MSA.

Here is a lookup table for magnitude differences:
Ratio Magnitude difference (appr) Ratio Magnitude difference (appr) 
0.4 1  2.5 -1 
0.5 0.75  4 -1.5 
0.64 0.5  6.4 -2 
0.8 0.25  10 -2.5 
1 0  16 -3 
1.25 -0.25  25 -3.5 
1.6 -0.5  40 -4 
2 -0.75  100 -5 

Blackwell's data, as used by Clark and me

Clark relied heavily on data from Blackwell to derive a theoretical framework, with concepts such as Optimum Magnified Visual Angle, Optimum Detection Magnification and many others. Blackwell and Clark had used data in the form of B0 (background brightness), BS (stimulus brightness) and C=(BS-B0)/B0 (contrast). I thought it would be more appropriate for amateur astronomy to see the brightness at the object's place as the sum of the contributions from the "background" and from the object itself. That is, the brightness of the object itself is BS-B0, and its total magnitude is this multiplied by its angular size. This is (more or less) known, and independent of magnification.

I have done some simple linear interpolation and derived threshold magnitudes as a function of object angular size, with different contrast levels as parameters. I have expressed contrast as -2.5*log10((BS-B0)/B0). Contrast=0 means that the object brightness and background brightness are equal, and added. Contrast=5 means that object is 1% of background, so you see an area of 101% of the background brightness. For contrasts less than 0, the object is brighter than the background.

Some points regarding this diagram:

The diagram shows what you can see in a hypothetical "zoom binocular" of 7 mm aperture, varying the magnification from 1 upwards. In a real telescope, you must add the aperture gain, and adjust for optical losses.

Following a line to the right means increasing the magnification, and decreasing the background brightness. For high contrast objects, the threshold rises steeply up to about magnitude 6, the maximum threshold for direct vision. When the background darkens from about 19-20 MSA to 24-25 MSA or even darker, averted vision is more sensitive and the threshold increases further toward magnitude 8. However, when the object is magnified above 1 degree apparent angular size, increasing magnification further leads to a decrease in threshold. For contrasts of 1.5-2 there are two maxima, one near 10 arcmin when you use direct vision, and one at1-2 degrees when you use averted vision. For lower contrasts you will see best with direct vision, and here the maximum threshold is near 10 arcmin apparent size.

To my surprise, my results did not quite match Clark's. The "optimum detection magnification" that I found was rather less than Clark's, and not so well defined. A bit of analysis made the reason clear (at least to me): on page 11 he writes "…the optimum magnified visual angle occurs when the first derivative (the slope) of each curve in Figure 2.6 is equal to -1." First, the curves are for constant background, not contrast, and thus do not represent changes in magnification. Second, had they done so, the slope should have been -2, not -1. This double fault leads to an apparent "maximum" that represents something else, but is not so very far from the truth.

Unfortunately, this is a very human mistake, but it invalidates most of Clark's concepts. This is all the more sad since he has written a great book, and most of his advice is very good indeed (I did write to dr Clark in 1991, and got a very friendly reply "explaining" my errors. I wrote a second letter but did not get a reply).

To keep it simple….

The essential message from the diagram can be memorized as a rule of thumb:

To detect a faint object, you can increase magnification till the sky is so dark that you have difficulty seeing the field stop, or till the object has an apparent size of 1 degree, whichever comes first.

However, note that published photographic object sizes are often much larger than the "core" from which most of the light comes, and you may use more magnification than a simple calculation may show.

I believe the increased light sensitivity with averted vision possible at high powers is a function of better dark adaptation, and thus takes time. The best dark country sky (22 MSA according to Brian Skiff, 24 MSA according to Clark) is too bright to allow full adaptation. So, if you are well adapted to this level, a few minutes with your eye fully protected from light, or looking only through a high power eyepiece, will make all the difference! Eyes adapt individually, and you can have one eye well adapted, and use the other eye to look in the finder, at star charts etc.

One point that neither Clark nor Blackwell discusses is the effect of using only one eye. The data are for binocular vision, and if the total amount of light is what counts, the threshold would be 0.75 magnitudes lower for monocular vision. Maybe it is not so bad, and 0.4 magnitudes lower is more realistic, but I don't know.

The thresholds here are for catching barely visible faint objects. If an object is brighter than that, it may be possible to see detail by increasing the magnification even further. To get the most out of observing, you should (as Clark points out) use several different magnifications, including very high. Also, it is difficult to get the most detail with averted vision, and you should take very good time to study an object.

About Blackwell's paper

A few years later I came across Blackwell's paper in our University library. This research must have been part of the war efforts. It is concerned with determining visual threshold contrasts for test stimuli of varying angular size against varying background illuminations, and here I want to mention some points that I consider important but are not mentioned by Clark. I have converted the original levels in footlamberts to MSA, having derived the relation 1 footlambert=11.3 MSA.

The test persons, young women volunteers, were given time to adapt to the illumination, and the "threshold contrast was defined as the contrast which was detected with a probability of 50 percent, due allowance having been made for chance success." Blackwell also mentions that "…the observers did not feel confident of having 'seen' a stimulus unless the level of probability of detection were greater than 0.90.", and this occurs for a log contrast of about 0.2 (-0.5 magnitude difference) brighter than the threshold. The test persons were free to use direct or averted vision, but Blackwell notes that they tended to use averted vision when background levels were darker than 19.2 MSA, and the curves show a marked discontinuity here.

In one series, the test persons had 6 seconds (a very short time for this task) to locate the stimulus in one of 8 positions. In another series, the stimuli were darker than the background. In a last series, the test persons had 15 seconds (deemed sufficient) to decide whether or not a stimulus was present in a given place(data in table VII). The smoothed data from this series are tabulated in Blackwell's table VIII, from 3.8 to 23.8 MSA. It is from this table that Clark has "derived" data in his figure 2.6, apparently by fitting functions to the data. I have plotted the limiting integrated magnitude as a function of background brightness using both table VIII and Clark's table F.4 for comparison. Clark's data show some possibly artificial crowding of the lines around 20 MSA background and some not quite credible extrapolation effects around 24-27 MSA. The differences are not serious, and I have used Clark's data for simplicity in my first diagram.

Table of my interpolation values, used in the first diagram above:

First row shows object size in arcmin, first column contrast in magnitude differences (5 lowest, -5 highest!). The following column pairs show first threshold magnitude, second background in MSA (bold is for backgrounds fainter than 19.2 MSA, where averted vision is preferred).
.595 3.6 9.68  18.2 55.2 121  360
5 3.15  11.71 3.37 13.30  2.24 14.58 1.20  15.24 -0.40 16.01 
4.5 -2.7 4.19 4.04  13.10 3.92 14.35  2.81 15.65 1.83  16.37 0.03 16.94 
4 0.94 8.35 4.57  14.13 4.32 15.25  3.23 16.57 2.33  17.38 0.94 18.35 
3.5 2.70 10.61 4.93  14.99 4.64 16.07  3.59 17.43 2.86  18.39 1.78 19.69 
3 3.75 12.16 5.17  15.73 4.88 16.81  3.98 18.31 3.54  19.59 2.94 21.35
2.5 4.46 13.37 5.36  16.42 5.08 17.51  4.48 19.32 4.43 20.97 3.88  22.79
2 4.98 14.39 5.52  17.07 5,26 18.18  5.22 20.56 5.34 22.39 4.74  24.15
1.5 5.34 15.25 5.62  17.68 5.41 18.84  5.94 21.78 6.04 23.58 5.26  25.17
1 5.57 15.98 5.70  18.26 5.73 19.66  6.60 22.94 6.51 24.56 5.62  26.03
0.5 -.03  6.97 5.64 16.55  5.75 18.81 6.23  20.66 7.18 24.01 6.86 25.40  5.82 26.73
0 1.96  9.46 5.70 17.11  5.83 19.39 6.88 21.81 7.58  24.92 7.13 26.17
-0.5 3.30  11.30 5.71 17.62  5.92 19.98 7.33 22.76 7.84  25.68 7.32 26.86
-1 4.16  12.66 5.75 18.16  6.72 21.28 7.68 23.61 7.97  26.31
-1.5 4.74  13.74 5.82 18.73  7.27 22.28 7.97 24.40 8.02  26.86
-2 5.15  14.65 5.86 19.27  7.55 23.11 8.18 25.11
-2.5 5.43  15.43 5.87 19.78  7.81 23.87 8.22 25.65
-3 5.63  16.14 6.53 20.94  8.01 24.57 8.22 26.15
-3.5 5.74  16.75 7.17 22.08  8.16 25.22 8.16 26.59
-4 5.81  17.32 7.54 22.95  8.24 25.80
-4.5 5.85  17.85 7.74 23.65  8.23 26.28
-5 5.85  18.35 7.86 24.27  8.16 26.72

Nils Olof Carlin - - comments are welcome

July 8th, 1997