In this paper, Schaefer presents a formula for predicting telescopic visual limiting magnitudes for stars, using physiology data from previous literature. To validate his formula, he uses over 300 observations by several observers.

In my discussion here, I present Schaefer's ideas as well as my own comments. I do it very much in my own way and order, trying to present them in a format that is hopefully easier to digest. One obstacle to the reader of Schaefer's paper is the somewhat odd mixture of units in his sources - e.g. star brightness in footcandles. I have converted (to the best of my ability) to magnitudes, magnitudes per square arcsecond (MSA - I hope you forgive me this non-standard abbreviation), and millimeters (you may -of course- go from inches to millimeters by multiplying by 25.4). This means that instead of multiplying many factors, I have chosen to add terms of magnitude.

Schaefer gives formulae both for day (direct) and night (averted) vision, but I will only consider the latter (as he does when validating his formula).

This is a logarithmic measure of star brightness, with roots back to the ancient Greek who divided the stars into six brightness classes. Pogson found that a logarithmic scale of 2.5 times the log(base=10) of brightness ratio would fit the old classes reasonably well. A brightness ratio of 100 to 1 thus corresponds to a difference of 5 magnitudes. The scale is "backwards", in the sense that fainter stars have higher magnitudes - this may be a stumbling block to the unwary.

Since stars have different "surface" temperatures, they will appear of slightly different colors. To measure this, astronomers use photometers with filters. The "visual" magnitude, V, is measured with a filter with peak transmission at 550 nm (5500 Å), matching the combined sensitivity of the cones of the eye that are responsible for day vision. The "blue" magnitude, B, matches the traditional blue-sensitive film emulsions with a peak sensitivity at 440 nm. The sensitivities are matched to make B=V for Vega, a bluish star.

For stars of other colors, the B and V magnitudes differ, and the difference (B-V) is known as the color index. It is larger for red stars, e.g. for Betelgeuse it is 1.86.

The rods responsible for night (averted) vision have their peak sensitivity at about 500 nm. They are more blue sensitive than the cones, and blue stars will seem brighter than their V magnitudes would suggest. Schaefer claims the color correction is "adequately represented" by taking the average of B and V magnitudes, this translates to a color compensation term (13):

This means that a blue A0 star with (B-V)=0 will appear a full magnitude brighter by averted vision

The major factors determining the visibility of an object star are the apparent brightness Ia of the star, and the apparent surface brightness Ba of the background it is seen against. Both these factors are modified by the telescope used, in different ways. The apparent star brightness is determined mainly by the area of the aperture, and the light collected will be "squeezed" into a thin pencil beam (the exit pupil) going into your eye.

The apparent background surface brightness, on the other hand, is never larger than seen with naked eye. With the use of high magnifications, the light available will be spread out to make a much darker background, against which faint stars are more easily seen.

The formula for visual limiting magnitude (from Knoll et al, converted from equations 2, 16 and 17) is:

This means that with a totally dark background, the limiting magnitude is 7.93. Here is a table of this equation for various backgrounds (darker than 18.4 MSA, where averted vision is used - Blackwell claims 19.2 MSA for this limit). Here, I have also listed the data from Blackwell using Clark's inter/extrapolations for the smallest test objects.

As Schaefer points out, Blackwell does not report the color temperature of the light sources. But he used incandescent lights whose color temperature must reasonably be close to Knoll's.

Given the visual limit, the apparent background brightness Ba can be had from the inverse of the formula above:

The data above are for limiting magnitudes for binocular vision. If only one eye is used, this lowers the sensitivity by a factor of 1/sqrt(2), i.e. a loss of sensitivity of 0.38 magnitudes:

The size of the eye pupil is important in Schaefer's estimates (it is fully dilated at the light levels of interest!). Schaefer gives equation (5) for average diameters (A=age):

The individual variations are large, however, and you would get a more accurate estimate by measuring the actual pupil size of the observer.

The eye is less sensitive to light entering the outer part of the pupil, but for night vision this Stiles-Crawford effect is small, and may be ignored. The formula used is (9):

For a 7 mm eye pupil, the light loss amounts to 0.12 magnitudes, for a 6 mm pupil it is 0.06 magnitudes. For smaller eye pupils or exit pupils, the light loss is negligible. For a simple approximation, if the eye pupil or exit pupil is less than 6 mm, set [Stiles-Crawford term]= -0.12, otherwise ignore (see discussion below).

To make the model fit the observed data, Schaefer had to introduce an individual sensitivity term, negative for an observer with more than average visual acuity (as great as -2.3 in one example), and positive for less than average acuity.

When validating his formula with observational data, Schaefer found a correlation between observing experience e and limiting magnitude. On a scale of e from 1 (beginner) to 9, Schaefer adds a correction term for this (eq 20):

Among other unmodeled effects, Schaefer mentions that hyperventilation may improve the limiting magnitude by some 0.3 mag.

The telescope acts as a light collector, compressing the light entering the aperture into a smaller bundle leaving the eyepiece. The diameter of this bundle is the exit pupil, and its diameter can be calculated as D[aperture]/M, aperture diameter divided by magnification.

The total light gathered is proportional to the increase in light collecting aperture over the naked eye, (D[aperture]/D[eye pupil])². This is represented by a negative term,

As long as the exit pupil is smaller than the eye pupil, all of the available outgoing light can enter the eye pupil. If the exit pupil is larger, some light is blocked by the iris, and the light loss is (eq 6) (this term must not be negative):

This commonly happens when people with less than 7 mm eye pupils use low-power binoculars such as 7x50, with their 7.1 mm exit pupils. Telescopes with selectable magnifications will normally be used with exit pupils smaller than the observer's eye pupil.

Using extreme magnifications, the star image may be magnified to appear as an extended source, and this will affect the visibility. If the seeing disk is magnified to an apparent angle of more than 900 arcsec (15 arcmin, or half the apparent size of the moon seen with naked eye), this calls for a correction term (eq (7), d = diameter of blur, in arcsec):

In perfect seeing, the apparent diameter of the Airy disk (or rather the dark ring ) is 2.44*M*lambda/D[aperture], and for lambda = 500 nm the apparent diameter is 15 arcmin or more at a D[exit pupil] of 0.28 mm or less. In poorer seeing, particularly with large apertures, this term is significant for larger exit pupils.

There will be unavoidable light losses in the optical components of the telescope. An uncoated glass surface will cause a loss of around 4% (=0.044 mag), and coating reduces it to 1% (0.01 mag) or less. A standard aluminized mirror loses about 12% per reflection (=0.14 mag), and with enhanced coatings this can be improved to 5%(=0.06 mag). Dirty surfaces will, of course, increase losses. Schaefer estimates this by judging the cleanliness from 1 (dirty) to 9 (freshly cleaned), subtracting 1% of transmission (adding 1% of loss) for each unit below 9.

A secondary obstruction will also cause some light loss. Schaefer assumes 15% obstructions by diameter (=0.025 mag) for all instruments. I believe 33% is more typical of SCTs (=0.13 mag) and 20% for Newtonians (=0.04 mag).

A Newtonian with standard coatings can have 0.32 mag , with enhanced coatings 0.15 mag loss with clean optics. A SCT could typically have 0.5 mag loss with uncoated optics, and 0.24 with best coatings. In addition to this, the eyepiece losses would be 0.04 mags for coated eyepieces such as Ploessls or Orthoscopics, up to 0.08 for more complex modern types such as Naglers.

For O'Meara's 24" reflector, Schaefer's estimate of total losses is 0.30 mag.

Even pure air will scatter and absorb light. Starlight from zenith will pass one "air mass" on its way to sea level, and the extinction will be 0.2 visual magnitudes in "good conditions". In "more typical weather" Schaefer claims a loss of 0.3 mag. The loss can, of course, be a lot more than this when there are more aerosols, and less when you are above a large part of the atmosphere, like on Mauna Kea. In his estimates, Schaefer uses measured the average extinction within 200 miles of the actual observing location, but doesn't elaborate on that. (Star magnitude estimates are given as compensated for extinction, i.e. as seen above the atmosphere).

For shorter wavelengths, the absorption is a bit more, so for averted vision, multiply this by 1.2.

The extinction is proportional to the secant of the zenithal distance - the lower the altitude, the more extinction. To compensate for this, I suggest you multiply the extinction at zenith by:

1.2 for altitudes of 50-65 degrees,
1.5 for altitudes of 35-50 degrees,
2.0 for altitudes of 25-35 degrees,

The sky background brightness can be expressed in magnitudes/square arcsecond (MSA). The background brightness comes largely from sunlight scattered by dust within the solar system, with contributions from skyglow, that is from ionized gas in the upper atmosphere, and also from the unresolved star background from the Milky Way. On top of this comes light from man-made sources, scattered in the atmosphere, and this can be the major source near cities - this light pollution can be highly directional.

The best observatory sites have backgrounds at zenith of 21.8 (visual) MSA, and a good country sky can have about 21.0 MSA (this means the brightness of one mag 21 star per square arcsecond, with the light evenly spread out!)

To allow for the different color sensitivity of night vision, a correction must also be applied to the sky background. A moonless night sky has a color index of 0.7, which gives a term of

(Light pollution from high-pressure sodium lamps will be redder, and if you can obtain a visual brightness for this situation, [Color compensation] will be higher - i.e. less negative.)

Lacking data on sky brightness, you may estimate it by finding the faintest star near zenith with your unaided eye(s), and then calculate "backwards". Here, most of the terms are zero (Schaefer sets [Stiles-Crawford term]=0) including [experience], but atmospheric loss and color compensation are important. The calculations are outlined below.

The background, as seen in a telescope, will be "spread out" by the magnification, and appear progressively darker with higher magnifications. This is the reason why higher magnifications can improve the limiting magnitude. This goes the other way, but only so far: if the magnification is low enough to make the exit pupil larger than the eye pupil, some light is lost and (but for optical losses) the brightness is as seen with the naked eye. Thus we will apply a correction (Schaeferīs factors Fm, Fa and Fp combined - here I take a mathematically equivalent but hopefully simpler approach). If the exit pupil, D[exit pupil]=D/M, is smaller than the eye pupil (note this term should never be negative):

Starting with your best estimate of the sky brightness B at zenith, you add the following terms to arrive at the apparent sky brightness:

This done, you can either start with the star magnitude, add the relevant terms and see if you reach the visual limiting magnitude I for this background, or start with the visual limiting magnitude and subtract the terms to get the magnitude of the faintest star visible. Here is the first approach:

If Ia is less than or equal to I, for the estimated Ba, the formula predicts that the star is visible.

To use the naked eye limit at zenith to estimate the background, you can use the expression above but with several terms being zero (see also discussion). If Is is the known visual magnitude of the faintest star seen, its apparent brightness is:

Insert in the inverse formula Ba = 21.58 - 5 log(10^(1.586-I/5)-1) or use tables, and apply corrections to get back to B:

Discussion

The Schaefer approach, with empirical corrections, can predict the visual limiting magnitudes, typically to within 0.5 magnitude and usually to within 1 magnitude. There will always be some experimental error, but several of the estimates involved cannot be made with very high precision by amateurs. In particular, the atmospheric extinction can vary greatly from one night to another, with moisture or dust from various sources. The sky background can vary with light pollution, of course, but also with the skyglow intensity (I think the sky brightness can be measured reasonably well with simple, home-made equipment, but that's an altogether different story).

Nevertheless, the calculations will clarify the importance of many factors involved.

The formula for the limiting magnitude I (Knoll et al) used by Schaefer gives a considerably steeper curve than the Blackwell/Clark data. If the curve is too steep, it may lead to an overestimate of the gains in limiting magnitude to be had by using very high magnifications to darken the background.

Also, the Blackwell data apply to extended objects, not only stars. It seems likely that with the appropriate corrections of Schaefer, particularly the color compensation, the Blackwell data would give very good estimates of visual limits. One difficulty is that many deep sky objects have a very uneven brightness distribution, and it is not obvious how to estimate their "visual" size. Here is certainly room for more work.

Color compensation: Given the photometric measurements are done with light sources corresponding to (B-V)=2, the compensated magnitude should be of the form V+x(B-V-2). (Brian Skiff has mentioned work by Richard Stanton, in the Journal of the AAVSO, where x is visually estimated to be in the range of 0.1-0.3 [pers.comm.], somewhat less than Schaefer's x=0.5.)

Also, if the eye pupil is smaller than the exit pupil or when observing with the naked eye,

[Magnification dimming]=5 log (7/D[eye pupil]), but if the exit pupil is smaller,

When estimating the sky brightness from naked-eye limiting magnitude, you should also apply a correction term [eye pupil loss]= 5 log (7/D[eye pupil]). Add this term for I, and subtract it for B (it will be negative for pupils larger than 7 mm!).

Schaefer also calculates the [Stiles-Crawford term] using the age-corrected eye pupil, but by the same reasoning, it seems more appropriate to use D[eye pupil]=7 mm here, too.

Re [Binocular term]: Schaefer applies the binocular correction to the star brightness, but also to the background brightness. Against a background of 19 MSA, the binocular limiting magnitude is 4.77, and without background correction the monocular limiting magnitude is 4.39 - with binocular term added to make the background 19.38, the monocular limiting magnitude is 4.68, only 0.09 loss over binocular limit. Thus to me, it seems more reasonable not to apply the binocular term to the background.

One effect that may be important is that with increasing age, the eye lens turns more yellow. This will affect night vision more than day vision, and tend to increase the color correction term (i.e. make it less negative!)

There are many reasons that skilled experienced observers can reach fainter limits than beginners. Knowing from experience what a faint object looks like is obviously important. So is knowing where and how to direct the look for most efficient use of averted vision. Details of observing technique may make a difference - Schaefer mentions that going from 6 to 60 seconds observing time increases the limiting magnitude by half a magnitude, but I fear even this is far too little to catch the fleeting glimpses of photons randomly clustering to reach visibility for short, sparse moments.

The preparation must be important, such as shielding the eyes from sharp sunlight during the day, and giving ample time in darkness to reach maximum dark adaptation - even looking at a "dark" sky to estimate naked eye visibility will affect the dark adaptation considerably for some time. Keeping in good physical shape, and avoiding toxic substances such as tobacco and alcohol, must also be important. Foodstuffs like bilberries and carrots in moderation are not likely to do any harm, at least - but megadoses of vitamin A will. While low blood sugar levels will lower light sensitivity, trying to raise the levels by ingesting quickly resorbed sugars is more likely to harm by rebound effects from increased insulin levels.

In a posting to the newsgroup sci.astro.amateur, Jeff Medkeff claimed to have observed a 14.68 mag. star with a 4.5" Newtonian, from Coronado Peak at about 6000 ft. This was a very determined effort at seeing the faintest star possible. The claim was contested in other postings, but is it reasonable?

Jeff mentioned an observed 6.8 mag. visual limit, we can use this to estimate the apparent sky background (the altitude of the stars were not mentioned, but we assume they were high enough not to warrant any correction of background or air extinction):

From this, the apparent background brightness Ba can be calculated as 21.34 (or taken by interpolation from the table). The visual background brightness can be calculated:

This is slightly darker than the 21.8 of the best observing sites, but not unreasonable. A small error here will give a very much smaller error in the end result, as we can see when we calculate the apparent background brightness in the telescope. The aperture is 114 mm, and at 110x the exit pupil is 1.0 mm. The transmission losses are guessed here: primary mirror 0.12, secondary reflection 0.04 + obstruction 0.04, eyepiece 0.04, and clean optics.

From the formula or from the table, the limiting magnitude of Knoll/Schaefer is 7.67 (of Blackwell/Clark 7.96). Jeff's observation was done after between 2 and 3 hours of strict darkness, with a small field eyepiece and with a dark hood keeping out extraneous light - the telescope was aimed at the star field by an assistant. We may presume his dark adaptation is at least as good as, and likely better than, the standard. Let us calculate the apparent magnitude of his 14.68 visual magnitude star:

This is 0.46 magnitudes short of the Knoll/Schaefer limit (and 0.19 short of the Blackwell/Clark limit), but well within the errors of Schaefer's model.

Oct, 1998