Part 3: The Astronomical Image
This page introduces the optical principles necessary to understand the design and performance of astronomical telescope systems the telescope and eyepiece used as a visual instrument with the eye included as a third component. It is one of series: previous pages explained basic optics and telescope & eyepiece combined; subsequent pages discuss optical aberrations, eyepiece designs and evaluating eyepieces.
Included at the end of each page is a list of Further Reading that identifies the sources used and points to additional information available online.
Collimation (pronounced kôl-ǝ-mā-shŭn) is the condition in which the three optical axes of the objective, eyepiece and observing eye are exactly aligned and coincident. It is the first of two critical adjustments necessary to obtain the peak optical performance from a telescope. Miscollimation arises when any of the three optical axes is out of alignment with the other two (diagram, below).
Observing through a spherical symmetrical lens from an off axis position has the same optical effect as observing on axis through a distorted or asymmetrical lens. Miscollimation therefore produces aberrations in "on axis" or centered star images where optical images are usually flawless. These aberrations most often resemble coma or astigmatism. They are easiest to produce by moving your head from side to side while viewing a star centered in the field of a wide angle eyepiece or in a small focal ratio objective.
Objective Collimation. Collimation usually defined as the procedure of aligning the optical axes of the objective lenses or mirrors with the center of the eyepiece focuser. This is the most common source of mechanical miscollimation and also the type that is easiest to correct.
Adjustment knobs or screws are usually provided for this purpose in the mirror cells of the primary and secondary mirrors of a reflecting telescope. It is not normally possible to collimate the objective lenses of a refractor, or the corrector plate of a catadioptric (Schmidt Cassegrain, Maksutov Cassegrain) telescope. The users given this opportunity are usually not grateful for it.
The specific steps necessary to collimate a telescope vary with the telescope design and manufacturer. Nearly all commercially available telescopes are provided with illustrated collimation instructions, and instructions are available online.
A number of collimation aids are commercially available to assist the collimation procedure. The oldest and simplest is the Cheshire eyepiece (image, right) which is an open metal tube supporting centered crosshairs and capped at the observer end with a pinhole. The crosshairs are illuminated by a side opening, which is turned toward the sky or an observatory light. This "eyepiece" is placed in the focuser, and the crosshairs viewed through the pinhole; their intersection is intended to show the direction of the optical axis created by the focuser drawtube. The secondary mirror is aligned with the focuser when it is centered on the crosshairs; then the primary mirror is adjusted until the secondary mirror appears centered within it.
Laser collimation tools are available to assist collimation, and these are especially popular among the owners of large aperture, small focal ratio Dobsonian telescopes, where exact collimation is required and tolerances can be quite small. Current vendors include (among many) Baader Planetarium, Farpoint, HoTech, Howie Glatter and Catseye Collimation. These tools are generally effective only with Newtonian format telescopes (including Maksutov Newtonians), not Cassegrain format telescopes. HoTech offers a system that is specifically designed for Cassegrain telescopes, but it is expensive ($400 to $450 at this writing).
However, visual collimation is highly effective and entirely adequate in moderate to large focal ratio (/7 or more) telescopes, and the principles are the same in all cases where a star image is used as the collimation target.
(1) First a "rough" collimation is performed according to the manufacturer's directions so that the secondary mirror appears centered in the focuser opening and the primary mirror appears centered in the secondary mirror.
(2) A moderately bright star (magnitude 2 to 4) at a medium zenith angle (20° to 40°) is chosen as a target. In this orientation, the adjustment screws in either the primary or secondary mirror cells must be easily accessible. The star is centered in the field of a low power eyepiece and brought into defocus, with the eyepiece focal plane beyond (outside) the objective focal plane, in the extrafocal position. (SCT users without an eyepiece focuser should note the SCT focusing procedure, described in the next section.)
(3) The adjustment screws are used to center the secondary shadow within the circular star image. The choice of adjustments depends on whether the primary or secondary mirror is being moved and whether the star image is in extrafocal or intrafocal defocus. In all cases, the defocused star image will appear as a thick annulus or nested rings of light around the secondary shadow or a defocused star image. The rings will not be concentric, but crowded together on one side of the star image (diagram right, top #1), and the goal is to manipulate the star image until the rings are perfectly concentric (#2).
The trick is to think of the rings as forming a cone, whose axis of rotation is the optical axis (diagram right, bottom). The cone is tilted to one side, which brings the rings on the opposite side closer together. The adjustment knobs are used to shift the star image in the direction away from this "crowded" or brighter side of the star image. This causes the star image to move out of the center of the eyepiece field, and it must be recentered to evaluate the direction and amount of the next adjustment.
(4) When the low power collimation appears reasonably accurate, a medium power eyepiece is used to examine the defocused image and further adjustments are made, this time with a smaller (less defocused) star image.
(5) The medium power eyepiece is replaced by a high powered eyepiece, and collimation is adjusted further. At this point the out of focus star image will be so small that the secondary shadow is no longer visible. Instead the Poisson spot that appears at the center of the extrafocal star image is centered inside the star image circumference (diagram, left). This is often disrupted by atmospheric turbulence, but the concentric diffraction rings around the Poisson spot can be used to judge the collimation and determine the necessary adjustments.
(6) Finally, the focused image of the brightest available star is examined, with attention paid to the contraction of the image in the last millimeter of focusing, and to the tiny threadlike granules that appear in the flare or glare around the star image. These granules should appear to radiate away from the star image equally on all sides. Note that they will be quite sensitive to the location of the observer's eye pupil, which must be centered on the exit pupil. (This occurs when the eyepiece field edge is clearly visible on all sides.) If the eye pupil is centered, then the shimmering filaments can be used to judge the last refinements in the collimation.
(6) Most primary mirror cells are equipped with lock screws, which should be gently and gradually tightened in a repeated circular sequence until all lock screws are fixed but not firm (only contact and not pressure between lock screw and mirror cell is necessary). Tightening one screw all the way before moving to the next can disrupt the collimation.
Eyepiece Miscollimation. Miscollimation in the eyepiece is usually a more difficult problem to resolve. It can arise because the eyepiece is poorly manufactured (a very rare problem), the focuser is not centered on or perpendicular to the optical axis, or the clamping mechanism in the focuser drawtube is not symmetrical.
The alignment of the focuser can be checked with a Cheshire eyepiece, and most high end focusers have adjustment screws as part of the tube mounting plate. In cases of extreme misalignment, the mounting screw holes may need to be enlarged or redrilled and the focuser remounted on the telescope tube.
The clamping mechanism should always be used with eyepieces, despite the possible annoyance of the undercut, to maximize alignment with the drawtube and minimize variation in the position. The eyepiece should first be seated so that the shoulder is in contact with the rim of the drawtube. The clamp should consist of a thumbscrew that exerts pressure on a flexible metal band, seated inside an interior groove. To ensure even distribution of pressure, the long end of an allen wrench can be used to rotate the flexible metal band in its groove so that the gap in the band is located directly opposite the thumbscrew.
Alignment issues are exacerbated when a mirror diagonal is used. In a test of six mirror diagonals available to me I found errors of up to 30 arcminutes in the alignment of the mirror to the optical axis, and to my annoyance I discovered that the most expensive mirror diagonals were not the most accurate. (One was shimmed with a putty adhesive, the other with black tape.
Eye Misalignment. Correct positioning of the eye pupil around the exit pupil is a third source of collimation error. The variations in this error are shown in the diagram (below) as a side view of the relative position of the eyepiece (lens schematic), the eye, and the emerging pencils of light passing through the exit pupil (orange bar). Also shown is a schematic of the visual appearance of each aberration, with the always in focus contour of the eyepiece field stop indicated by dark gray, and the always strongly defocused contour of the exit pupil by a lighter gray.
In the correct alignment of the eye (diagram, top left), the observer's iris is placed so that it is in the same plane as and is centered around the exit pupil; the optical axis of the system passes through the center of the observer's cornea, lens and the fovea of the retina. The focused edge of the field stop is visible around the entire circumference of the image area, and any optical aberrations (astigmatism and apparent field curvature in particular) are radially balanced of equal size and appearance around the edges of the field.
Even in this position, it is possible for the observer by very small movements of the head to create skew in the spicules of scatter around bright stars or (in short focal ratio telescopes) comatic aberrations in stars at the center of the image area. These subtle cues can aid the observer in keeping correct eye collimation.
In longitudinal misalignment of the eye (diagram, bottom left), the observer's pupil is centered on the optical axis but is located in front of or (usually) behind the exit pupil. In either position, the exit pupil vignettes the image area and creates a "keyhole" narrowing of the field. If the target is a bright object such as the moon, a sufficient distance shows clearly the outline of the exit pupil in the center of the eyepiece eye lens.
In lateral misalignment (diagram, bottom right), the iris of the eye is displaced to one side of the optical axis; this causes a curved shadow or "shutter" (the edge of the exit pupil) to intrude into the field from the side of the eye's displacement.
Finally, axial misalignment (diagram, top right) is caused by the observer looking to one side of a wide apparent field while the pupil is approximately centered on the eye point: the fovea moves off the system optical axis. This does not produce a noticeable effect in traditional or well corrected wide field eyepieces, since the eye is always looking directly into an off axis but perfectly collimated "pencil" of light (blue rays in diagram). However, in wide or super wide field eyepieces with spherical aberration of the exit pupil, it is necessary to displace the eye in a direction opposite of the direction of view, which hides the field stop edge behind the exit pupil. In addition, a floating lenticular or "kidney bean" shadow appears in the field on that side, which is often outlined by the smeared out light of stars.
While it is a matter of physical skill to find and hold the correct eye position while observing, various supports can make this task enormously easier. In particular, an adjustable observing chair is very convenient for observing with most refractors and Cassegrain format reflectors; Dobsonian and Newtonian reflectors should be used if necessary with sturdy, stable ladders or stepladders, preferably with a raised front rail that allows the observer to steady their upright posture at the eyepiece. No matter what kind of telescope is used, positions that require bending or leaning, or strain in the neck, back or legs should always be avoided as much as possible.
Focus is the second of two critical adjustments necessary to obtain the peak optical performance from a telescope. By design both the objective and eyepiece have a specific focal surface; focus is the distance adjustment along the optical axis that brings the two into coincident position.
The objective focal surface is the point of reference:
The area on the objective side of the objective focal plane is called intrafocal or "inside the focus". Moving the eyepiece out of focus in this direction causes the output rays to converge, corresponding to an increasing positive diopter setting that is appropriate for eyes that undercorrect (are farsighted).
The area on the eyepiece side of the objective focal plane is called extrafocal or "outside the focus". Moving the eyepiece out of focus in this direction causes the output rays to diverge, corresponding to an increasing negative diopter setting that is appropriate for eyes that overcorrect (are nearsighted).
Focusing Mechanisms. In a typical rack and pinion (gear and linear toothed track) focuser or Crayford (roller and cylinder) focuser, the eyepiece is adjusted by a pair of knobs on a transverse shaft below the optical axis. The rotation of the knobs necessary to move the eyepiece in or out is immediately obvious from the visible movement of the eyepiece. Rotating the right hand knob clockwise moves the eyepiece toward the objective on the optical axis, in the intrafocal direction; counterclockwise rotation moves the eyepiece away from the objective.
In a commercial Schmidt Cassegrain system, in which the focus is adjusted by moving the hidden primary mirror using a mirror cell focusing knob protruding from the back of the telescope, the necessary adjustment is less obvious (diagram, right):
Turning the mirror cell focusing knob clockwise moves the primary mirror away from the secondary mirror, decreasing the focal length and moving the focal surface away from the observer. An eyepiece in focus is moved out of focus in the extrafocal direction, and an eyepiece out of focus on the intrafocal side is brought into focus.
Turning the mirror cell focusing knob counterclockwise moves the primary mirror toward the secondary mirror, increasing the focal length and moving the focal surface toward the observer. An eyepiece in focus is moved out of focus in the intrafocal direction, and an eyepiece out of focus on the extrafocal side is brought into focus.
Focusing Procedure. The optimal focusing procedure is always to start focus from an extrafocal position the eyepiece focal plane must be in front of the objective focal plane (diagram right, bottom). The reason has to do with how the eye focuses the diverging light rays from nearby objects. In the relaxed used to view distant objects the eye lens is actually stretched into a flattened figure by zonule fibers or ligaments anchored in the iris muscle. To focus on nearby objects, the iris muscle contracts in a way that relaxes the tension on these ligaments and allows the eye lens to bulge into a shorter focal ratio. Age hardens the lens, reducing and (around age 60) eliminating the elastic ability of the lens to respond to these changes in tension.
When an image is brought into focus from an intrafocal position, it approaches focus with a "nearsighted" convergence of rays. The eye handles these as it would an object near the eyes, and tightens the iris muscles in an attempt to focus; some of this tension remains even after the image is brought into focus. In contrast, an image brought into focus from the extrafocal direction approaches focus with a "farsighted" divergence of rays. The eye cannot accommodate toward the image any farther than a completely relaxed state, and this remains once the image is brought into focus. If you overshoot the correct focus and err onto the intrafocal side, return to the extrafocal position and start again.
In commercial Schmidt Cassegrain telescopes, changes in the primary mirror alignment ("mirror shift" or "mirror flop") occur when focusing with the mirror adjustment knob. The recommended focusing procedure is first to turn the focusing knob clockwise to bring the eyepiece into an extrafocal defocus, then to bring the eyepiece back into focus by a counterclockwise adjustment in the intrafocal direction. This should be the same focusing procedure used during collimation (except that the star is left somewhat out of focus on the extrafocal side), so that the mirror tilt during collimation and in routine observation is the same.
The ability to produce a sharp focus is partly dependent on the optical quality and collimation of the instrument and partly dependent on the amount of atmospheric turbulence. Nevertheless, the "best" focus will always produce the visually smallest or most compact image of a star, and the sharpest edge in a planetary disk or lunar terminator, when the target is placed at the center of the eyepiece field of view, and will make visible the greatest number of faint stars in the field. Under optimal adjustment and viewing conditions the image will appear to "snap" unambiguously into the best focus although the "snap" is most vivid in small focal ratio telescopes, and atmospheric turbulence can make this test unreliable.
When poor seeing makes focusing difficult, best focus is found by bracketing the focus. First bring the star into intrafocal defocus to a recognizable disk visual diameter, then take the star in the opposite direction, through the focal position into extrafocal defocus, until the disk reaches the same visual diameter. To finish, rotate the focusing knob back by half the travel between the two defocus positions.
Poor seeing can cause large variations in focus. Resist the temptation to continually adjust focus in order to "chase the seeing". The best procedure is not to make continual adjustments, but to find the focus that produces the most frequent intervals of focus. Patience, rather than manipulation, is the bast focusing procedure: focus should be changed by very small increments so that the quality of the new focus can be observed for several seconds.
Depth of focus. This is the distance in front of and behind the optimal focal point within which image quality is still acceptably sharp. If the standard of image quality is that it is not degraded by more than 1/4 wavelength of light (the Rayleigh criterion), then:
DF = ±2λNo2
Depth of focus is proportional to the square of the relative aperture. Thus an /20 Cassegrain telescope has a depth of focus of ±0.44 mm at λ = 550 nm, but an /4 Newtonian has only ±0.02 mm. These tolerances are especially critical for astrophotographers, who typically use automated (software driven) focusers to find the optimal focal point at the camera image plane.
(1) Geometry of Magnitude. The magnitude system was proposed in 1856 by Norman Pogson, who noticed that a five magnitude difference in the ancient system of naked eye magnitude ranks (attributed to Hipparchus and adopted by Ptolemy) corresponded very nearly to a 100 times difference in brightness. Therefore, the base value of the magnitude system was chosen as 1001/5 = 2.51189, so that 2.5125 = 100.
Magnitude is a exponential metric in which the magnitude units are actually the exponents applied to a constant base value that represents a fundamental proportion or ratio. The base value 2.51189 is a unit of illuminance ratio: if two stars differ in brightness by 1 magnitude, then one star emits roughly 2.5 times more light than the other. The numerical magnitude is the exponent of this unit, and by using fractional exponents (continuous magnitude values), the base value can be multiplied to express any brightness ratio.
This type of scale mimics the relationship in sensor dynamics between the change in the energy of a stimulus and the change in the sensor (perceptual) response, known as the WeberFechner Law. Many empirical scales that equate a physical stimulus a perceptual response, including the Richter Scale (earthquakes) and the decibel scale (sound), are exponential metrics.
Originally, Vega anchored the scale as a 0.0 magnitude star. A star at the naked eye limit magnitude of 6.5 has 1/2.5126.5 = ~1/400 the brightness of Vega. The system can be extended to objects of any illuminance by using negative values for objects brighter than 0.0 in other words, increasing magnitude denotes decreasing brightness.
The exponent character of the scale means that equal brightness ratios describe any equal magnitude difference. Thus, a 10th magnitude star is only 1/10,000 as bright as a 0 magnitude star such as Vega, and the same ratio applies between 2 and 8 magnitude stars.
Relative brightnesses, the aperture diameter required to grasp equal illuminance from fainter magnitudes, and the parsec distance of a star of 5.0 absolute magnitude (close to the Sun's 4.83 absolute magnitude), are listed in the table below down to v.mag. 10.
The values for brightness ratio and proportion define the magnitude system. The physical insights are in the aperture diameters and physical distances.
The aperture diameters represent multiples of any measurement unit (centimeters or inches); the difference between the anchor magnitude in the table and the limit magnitude of a real aperture can be calculated as a constant, then applied to any other aperture as the anchor magnitude plus the constant.
Let's use the naked eye pupil as an example. Its aperture diameter is approximately 0.6 cm, and in the table this is anchored at a magnitude of 1. Its actual limit magnitude is approximately 6.5, which is 7.5 units from the tabulated anchor value of 1. A 100 cm (1 meter) telescope is 100 times as large, and this ratio is equated in the table with a magnitude of 10, so its limit magnitude for visual astronomy will be 10 plus the constant 7.5, or 17.5. An exact calculation (formula , below) yields 17.6 for the limit magnitude of a 1 meter telescope. The same constant applies to any other magnitude: a 4th magnitude star viewed with the naked eye will appear as bright as a 11.5 magnitude star viewed with a 100 meter aperture.
If we use any aperture diameter as a benchmark, then the aperture diameters apply to multiples of that diameter. For example, if we equate an aperture of 1 cm with 0 magnitude, then we find it collects as much light from Vega as does a 100 cm aperture from a 10th magnitude star. If we equate a 4 inch aperture with 0 magnitude, then there is a gain of three magnitudes with a 16 inch aperture, four times as large. (See also the limit magnitude graph, below.)
The distance metrics are governed by the inverse square law, which states that moving a light source 2 times farther away will reduce its brightness to 1/4th (1/22). Thus, if a star is at a 1 parsec distance, then moving it to 10 parsecs will reduce its brightness by 1/102 = 1/100th, which the table shows will increase its magnitude by 5. Given two stars of equal absolute magnitude, one 7 times farther than the other, the farther star will be 1/72 = ~1/50 as bright. In the table the ratio 1/50 falls between magnitudes 4 and 5 but is much closer to 4, so the farther star will appear to be about 4.2 magnitudes fainter. (Exact calculation with formula  below yields a magnitude difference of 4.27.) The magnitude/distance relation means that stars as bright as the Sun (absolute magnitude ~5.0) disappear from naked eye view at around 20 parsecs. If Vega (7.7 parsecs distant) were 25 times farther away (at 192 parsecs) then it would appear to be a 7th magnitude star.
This discussion illustrates the constant proportions between magnitude and brightness (the constant 2.512), magnitude and distance (the inverse square law, used in the distance modulus), and magnitude and aperture.
(2) Magnitude and Brightness. Astronomical references provide standard formulas to convert between magnitude difference and brightness ratio. Given a magnitude difference m'm (where m is the magnitude of the brighter star, m' > m), to find the luminance ratio b/b' (where b is the luminance of the brighter star, b > b'):
 b/b' = 2.512(m' m), or b/b' = 100.4(m' m), or log(b/b') = 0.4(m' m)
Given a brightness ratio, to find the equivalent magnitude difference:
 m' m = 2.512·log(b/b')
Note that 0.4 is the rounded value of the exact reciprocal 1/2.51189 (0.39811). Note also that brightness change is always a ratio (greater than one) and magnitude change is always a difference (greater than zero), because division is done in an exponential scale by subtraction of exponents.
The brightness ratio across all naked eye stars (from Sirius at 1.46 to the 6.5 magnitude limit) is then:
b/b' = 2.5126.51.46 = 2.5127.96 = 1528
However averted vision is required to achieve the low end of this dynamic range. If we put the foveal threshold (stars visible with direct fixation) at around magnitude 4.5 under conditions of typical sky brightness, then the dynamic range of dark adapted foveal vision is at minimum 2.512(4.51.46) = 2.5125.96 = 242 (see the table, above). This two magnitude difference suggests that averted vision is about 6.3 times more sensitive to light than foveal vision.
In double stars imaged as a single light source, the brightness of the combined starlight is not the sum of the separate magnitudes. Instead, the incremental brightness (x) is calculated from m'm, and this is subtracted from the magnitude of the brighter star (m):
 x = log(1+(1/2.512m'm))/0.4, then M = m x
Plugging in some basic values, we find that the combined magnitude of two stars of nearly equal magnitude (m'm = ~0) increases by log(1+1/1)/0.4 = 0.75 magnitudes, and this is subtracted from the magnitude of the brighter star. The magnitude increase is x = log(1+(1/2.512))/0.4 = 0.36 magnitudes when the magnitude difference is 1.0, x = 0.16 when the magnitude difference is 2.0, and x = 0.07 when the difference is 3.0. (The human eye is usually insensitive to magnitude changes less than ~0.2.)
To anchor these relationships as measures of illuminance or brightness, the light gathering power of the telescope can be expressed in three ways: (1) as the increase in light gathered over the light that enters an unaided, dark adapted eye; (2) as the limit magnitude of stars visible with averted vision, both visual parameters; and (3) as the illuminance of the telescopic image on a film or CCD sensor, a photographic parameter.
(3) Aperture & Light Grasp. The light gathering power of the telescope is a function of its aperture area (Ao): as the aperture increases, more light enters the system and is concentrated into the usable image. For a refracting telescope this area is usually calculated as:
Ao = Do2
Because the secondary mirror in a reflecting telescope blocks a certain amount of light from reaching the primary mirror, its area (Ds2) must be subtracted from the area of the objective in order to get the effective aperture (Deff):
 Deff = sqrt(Do2 Ds2)
Thus, a 250 mm (10") telescope with a 50 mm (2") diagonal diameter has an effective aperture of 9.6" or 245 mm.
The relative light grasp of any two apertures is proportional to the square of their areas, and the increase in light grasp with aperture scales as the square of the aperture ratio:
 ΔAo = (D1/D2)2
(4) Telescopic Gain Over Naked Eye Brightness. The first criterion requires an estimate of the dark adapted pupil aperture. To measure: obtain a complete Allen wrench set. At night, once your eyes are fully dark adapted (roughly 1/2 hour after the last bright light exposure), close one eye and pass an allen wrench back and forth in front of the other eye. Hold the long shaft of the wrench vertically with the short end pointed in the direction you are looking. If the wrench fully occludes a bright star, causing it to "blink", then the wrench is larger than your pupil opening. Try different wrenches until you discover the largest wrench that dims but does not completely block the star image. Then the width of that wrench is the diameter of your dark adapted pupil (δ).
Then the measure of a telescope's light grasp in comparison to your naked eye is the ratio of the two areas:
 Lo = (Deff/δ)2t
where t is an estimate of the system's light transmission, which is 0.98 for each surface of aluminum high transmission catoptrics, 0.90 for aluminum standard catoptrics, and approximately 2% per centimeter of fully multicoated refractor or eyepiece glass. My dark adapted pupil is about 6 mm wide, so a 10" reflecting telescope with a 2" diagonal will gather at most (245/6)2 x 0.98 x 0.98 = ~1600 times more light than my dark adapted eye, not accounting for light absorption by the eyepiece.
Magnification is a critical qualification in estimates of the visual brightness of a telescopic image: the objective gathers light, but image magnification spreads a fixed density of light over a square visual area. Thus the ratio of telescopic to visual brightness becomes:
(5) Telescope Limit Magnitude. Because visual magnitude differences are a direct measure of luminance ratios (expressed in logarithms), the telescope/eye ratio implies a 10" telescope limit magnitude that is 1656 times fainter than the visual limit magnitude.
Using (as a first approximation) the standard brightness ratio formula  (where b' and m' refer to the fainter brightness and magnitude, respectively), and assuming a visual limit magnitude of 6.5, yields:
log(2452/62)/0.4 + 6.5 = m'
for a Deff = 245 mm (10") telescope limit magnitude of 14.56. This standard formula is more often cited in the following equivalent form:
 m'o = mne 5log(δ) + 5log(Deff)
This form shows that the three critical variables are effective aperture (Deff) and pupil aperture (δ), both in millimeters, and naked eye magnitude limit mne (a function of atmospheric seeing, transparency, diffusion and light pollution, which are not measured directly). Assuming a pupil aperture δ = 6 mm and a naked eye limit magnitude mne = 6.5, the formula simplifies to:
 m'o = 2.61 + 5log(Deff)
and this yields a value m'o = 14.56 identical to the value calculated using formula . (To adjust the formula to your pupil aperture and naked eye limit magnitude, use formula .)
Attempts have been made to refine this straightforward calculation by including more variables magnification, zenith angle, observer experience, optical quirks of the eye under the statistically dubious assumption that a greater number of inputs will increase accuracy.
The graph (right) shows estimated limit magnitude, as calculated by Bradley Schaefer (1990), across 6" to 16" apertures and 5 levels of magnification, under the condition that the faintest stars are viewed with averted vision. The predictions assume δ = 6.7 mm, a naked eye limit magnitude mne = 6.0, a telescope transmission of 80% and atmospheric transmission of 70%. The predicted values correspond reasonably well to the actual limit magnitudes in 314 observing reports from over four dozen observers. The main effects shown in the graph are that limit magnitude increases with aperture and with increasing magnification (from 30x to 400x), but that the incremental increases decline significantly as aperture or magnification increases.
Schaefer claims to include the effects of human dark adapted visual sensitivity, atmospheric absorption, observer age corrected pupil size, telescope transmission, magnification, scotopic eye transmission, star and sky color temperatures and observer experience. But the largest additional impact is from the sky brightness (naked eye limit magnitude, NELM). The blue curves assume a visual limit magnitude of 6.0; the single green curve shows the effect of reducing the visual limit to 4.0 at a magnification of 100x. Note that this increase in the limit magnitude of a 10" objective between a suburban and dark sky site (a, Δm = 1.25) is roughly twice as large as the increase between a 10" and 16" objective (b, Δm = 0.66)! This illustrates the significant benefit of a medium aperture but compact and portable field telescope for observers with access to dark sky sites. Note too that increasing magnification from 100x to 400x increases the limit magnitude of a 250 mm aperture to slightly more than the limit magnitude of a 400 mm aperture at 100x (b). This illustrates the large impact of magnification on limit magnitude.
Schaefer's complex model predicts m'o = 14.15 for a 10" aperture at 60x magnification. One can ask whether the difference of 0.09 magnitude between this model and the estimate of 14.06 given by the simple formula  (using the same values for δ, mne and Do) is worth the added complexity. Subsequent and more intricate proposals by Nils Olof Carlin and Chris Lord do not improve the validity of the methods by using different variables or assumptions, because the error in the actual observed magnitude across observers, due to factors omitted or impractical to measure, remains on the order of ±0.5 magnitude the difference between a 6" and 8" telescope.
Schaefer's model underestimates limit magnitudes with a ±0.75 average magnitude error, and he comments that "the limit of accuracy for my model is much larger than I would have hoped or expected." That is because his expectations were unrealistic. Limit magnitude is difficult to predict because many stellar, atmospheric, instrumental and observer variables can affect it, most of them inconvenient to measure with the accuracy necessary for reliable prediction. Worse, estimates of visual acuity are an exercise in predictive psychophysics, which has always proven difficult, even with carefully controlled visual stimuli, due to the very large differences in visual capabilities across individual observers.
My suggestion is to use the explicit, simple formula  to obtain a baseline estimate, then revise this value with the judgment of experience to correspond to specific applications or observing conditions. Schaefer actually demonstrates this approach at the end of his research paper, where he makes several ad hoc adjustments to his model in order to "predict" Steven O'Meara's remarkable visual acuity.
(6) Photographic illuminance. The photographic illuminance of the telescopic image increases with aperture and decreases with relative aperture, making it proportional to:
 Io = (Do/No)2
Relative Aperture (Focal Ratio)
The relative aperture (No) of a telescope is its focal length divided by the diameter of its entrance pupil or clear aperture (Do):
 No = o/Do
It is also known as the focal ratio or ratio. The name and notation invites erroneous interpretation as a fraction (i.e., /2 is "larger" than /6), although relative aperture is actually the quotient of a ratio. Thus a relative aperture of /2 is comparatively shorter or photographically faster than /6, which is longer or slower.
Relative aperture is not a consistent measure of any physical quantity other than the maximum field radius ρ produced by a telescope objective:
 ρ = 2arctan(1/2No) radians.
Relative aperture is sometimes referenced in the context of optical aberrations. Thus, coma is considered to be a significant aberration in Newtonian reflectors shorter than /5, and the spherical aberration in a spherical mirror or lens becomes negligible (by the Rayleigh criterion) when
N ≥ 3.4(Dcm)1/3 or ≥ 4.6(Din)1/3
Both depth of focus (defocus tolerance) and depth of field increase with relative aperture.
In photometric contexts, relative aperture expresses the proportion of unit illumination (aperture area) per unit magnification (focal length) produced by the telescope objective. However, these units are dimensionless, and comparatively useful only by holding one element constant. For example, as the relative aperture gets larger for a constant aperture, the quantity of light projected onto a unit area of the image plane decreases because the image is magnified. Equivalently, as the relative aperture gets larger for a constant focal length, the quantity of light projected into the unit image area decreases because a smaller aperture admits less light.
Relative aperture determines the ratio between eyepiece focal length and exit pupil diameter, independent of aperture or focal length. Thus, an /10 telescope will produce an exit pupil that is 1/10th the focal length of the eyepiece, regardless of the dimensions of the objective or the image magnification.
Because magnification is proportional to focal length, but light grasp is proportional to the square of aperture, the relative image illuminance produced by different relative apertures (different focal lengths for the same aperture, or different apertures for the same focal length) scales as the ratio of the relative apertures squared (chart, above).
The geometry of relative aperture is luminance (Lo), calculated as the product of image area and aperture solid angle, or as aperture solid angle when image area and luminous flux are held constant. Then the formula for image luminance reduces to:
 Lo = E/N2
where E is the illuminance of the source in lux. The table below shows the aperture angular diameter for each relative aperture, with its area in steradians and illuminance in proportion to the luminance at /2. The diagram (right) illustrates relative aperture as distance from a small circular aperture in a dark room: as the image surface is placed farther from the aperture, the light falling on its surface decreases. Note that /2 is near the lowest feasible aperture, /4 is useful for richest field telescopes, and /8 is the lower value for traditional Newtonian reflectors, achromat refractors, and Schmidt Cassegrains. (The case /250 is relevant to the discussion of exit pupil.)
Thus an /2 system has roughly six times the image illuminance of an /5 system ((5/2)2 = 2.52 = 6.25), an /7 system has half the illuminance, an /10 system one fourth, and so on.
The standard system of photographic stops doubles the area of the diaphragm stop (effective aperture) at each step, which means the stops increase the image illuminance by powers of the square root of 2 (1.414): 1, 1.414, 2, 2.83, 4, 5.65, 8, etc.
The resolution of an optical system is measured as the smallest angular width that the system can image at the focal surface. In astronomical systems this limit is determined by three things: the quality of the optics of the system, the turbulence or distortion introduced by the atmosphere, and the quantum wave properties of light. Only the last property can be calculated directly as the reciprocal of aperture, 1/Do.
The Diffraction Artifact. Calculation of the aperture resolution limit depends on a quantum (physical) rather than geometrical analysis of light.
If the light wavefronts from a distant star are traveling in direction x parallel to the optical axis of a telescope, then the width of each wavefront is measured along a dimension y perpendicular to the direction of light. Because the star is so far away, the scale of the y dimension is effectively too large to measure.
The quantum indeterminacy of the location and momentum of any photon in the wavefront is distributed across the entire dimension y, and both the energy and location of the photon can have a precise value, which means it can carry information about the emitting source of the photons.
In a telescope, the wavefront location for each photon is fixed at the focal surface of the objective. But because the wavefront has passed through the aperture stop, the indeterminacy distributed across the entire y dimension is reduced to the aperture width D. This introduces a quantum uncertainty or "error" in the specific location of each photon in the image, which must be no smaller than:
U = hν
where h is Planck's constant, 6.626 x 10-34, and ν is the frequency (energy) of the light, and where the speed of light c is their product, c = νλ. Casting these quantum relationships in a form that defines the indeterminacy of a circular aperture of diameter D yields:
 UD = hc / hνD = hνλ / hνD = λ / D
If we specify the energy of light by wavelength rather than frequency, for example as the commonly used λ = 550 nm (approximately the eye's peak photopic sensitivity), then the angular width of the indeterminacy in a 10" (254 mm) telescope becomes:
UD = 0.00055 / 254 = 2.165 x 10-6 radians
In the light from a far distant star, the result of the cumulative uncertainty from all photons focused to a single image point does not have a simple "fuzzy" image structure around the point, due to the destruction or reinforcement of light waves as different parts of the wavefront are superimposed at the same location on the image plane. Instead it appears as a central concentration of light surrounded by one or more rings, the center and rings separated by concentric dark intervals.
The central concentration of this diffraction artifact is called the Airy disk (after the Cambridge professor George Biddell Airy who first described it mathematically in 1834), surrounded by concentric diffraction rings. The angular dimension of these features, in relative units of λ/D, the relative luminance (as a proportion of the peak Airy disk luminance) and the percentage of total light energy enclosed by each dark ring are given in the table (below).
Angular Resolution Criteria. This diffraction artifact the Airy disk and encircling rings is the smallest possible physical image of a "point" light source. As a result, the diffraction artifacts created by two "incoherent" or physically unrelated light sources (such as two stars) will appear to merge on the image plane when the angular separation between them is smaller than the angular diameter of the artifacts.
What is the minimum angular separation between two point sources necessary to resolve their overlapping diffraction artifacts? This is the angular resolution criterion (Ro) for a given aperture. It is determined by the wavelength of light taken as the reference, the unit of the diffraction artifact used as the minimum resolvable detail of the image, and the aperture diameter:
 Ro = k·λ/Do·206265 arcseconds
where λ and Do are measured in millimeters, and the constant 206265 converts the tangent (radians) into arcseconds.
The factor k constitutes the definition of what optical or visual "resolution" means in relation to the fundamental λ/Do metric. Since the late 19th century a variety of standards have been proposed and used as resolution criteria; the four most often encountered in astronomy are:
Note the near equality of the λ/D and Dawes criteria: for example, the minimum interval resolved in a 12 inch (305 mm) aperture at the λ/Do criterion is Ro = 113.4"/305 = 0.372"; with the Dawes criterion, it is 0.379". The diagram (below) illustrates the visual implication of various resolution limits with FWHM Airy disks.
Angular Telescope Magnification. Once the angular size of the smallest resolvable interval is calculated for a specific aperture, the practical question is: how much magnification is necessary for the observer to resolve it visually?
Assuming that the eye and telescope are both diffraction limited, a good rule of thumb is that the required magnification will equal the aperture diameter in millimeters: a 305 mm aperture will require about 305x magnification.
A more accurate answer requires a measure or estimate the observer's angular visual resolution limit (Rv). This is usually stated as 120 arcseconds for normal vision. Then the necessary magnification is:
[5a] Mt = Rv/Ro
[5b] e = o / (Rv/Ro)
Given the standard angular visual resolution limit Rv = 120", the required magnification in a 305 mm aperture for the λ/D criterion would be Mt = 120"/0.372" = 323x. My visual limit is about Rv = 100", which requires only about 270x. This large range in the estimated necessary magnification is typical of the differences in visual acuity across individual observers.
Linear Resolution Criterion. Resolution can also be analyzed in the dimensions of the image plane, where the image is at much too small a scale to be utilized by naked eye. The function of the eyepiece is to provide the dioptric magnification necessary to make image details visible. This requires comparison with the observer's linear visual resolution limit (Rv) or, without the eyepiece in astrophotography, the width of individual pixels in a CCD receptor.
Unlike the angular resolution in image space, which scales only with aperture Do, the linear or physical width of the diffraction artifact on the image plane (So) scales only with relative aperture No, independent of aperture and focal length:
 So = k·λ·No millimeters
where k is the scaling factor of a resolution criterion based on a diffraction feature (see formulas 3a-3d, above). Given an /8 telescope and λ = 0.00055 mm, the linear diameter of the FWHM Airy disk is 2.03 x 0.00055 x 8 = 0.009 mm about the size of a single red blood cell. The eyepiece enlarges this small interval so that it is visible to the eye.
Formula  is the quickest way to derive the linear resolution limit. For reference, the table below shows the linear metric (at No = 1) and the matching naked eye linear criterion (at No = 250) corresponding to the four angular resolution criteria (for λ = 0.00055).
The linear metric must be multiplied by No (or the matching naked eye criterion by the quotient of No/250) to obtain the resolution limit for a specific relative aperture. For example, in an /10 telescope with the λ/D criterion, the linear resolution limit is either So = (10/250)*0.1375 = 0.04*0.1375 = 0.0055 mm, or So = 0.00055*10. (See also the table below.)
Linear Eyepiece Magnification. How much eyepiece magnification is necessary? As before, that depends on the observer's visual acuity, defined by a linear visual resolution limit (Sv). This is measured as line pairs per millimeter (lp/mm) or as the spacing between parallel white and black lines viewed at the near point (250 mm) with the naked eye. The typical value is 6 to 8 lp/mm or an average line spacing of about 0.1429 mm.
In the example of an /8 objective given above, we'd require an eyepiece magnification of 0.1429/0.009 = 15.9x to barely detect the FWHM Airy disc; the formula for dioptric magnification indicates this is roughly a 250/15.9 = 16 mm eyepiece.
The formula to calculate directly the eyepiece focal length (e) at which the diffraction limited image details will be just visible is:
 e = 250·(So/Sv) millimeters
Thus the eyepiece focal length that will resolve the smallest image detail under the λ/D criterion in an /8 telescope (So = 0.0044 mm), given a visual resolution of Sv = 0.1429 mm, is
e = 250·0.0044/0.1429 = 7.7 mm
which is an eyepiece magnification Me = 250/7.7 = 32.5x. (See also the comparative table here.)
Line Resolution Criterion. In an afocal system with circular aperture, the unweighted quantum formula is used to compute the minimum resolvable interval between two lines is:
 νo = λ/Ds·206265 arcseconds (interference rings or bands)
As already explained, this is equivalent to the spacing between dark rings of the point diffraction artifact or the dark bands produced by a two slit aperture mask, where Ds is the distance between the centers of the two slits.
Choice of Resolution Criterion. Because these are practical resolution criteria, and implement different conceptions of what resolution means, none of them should be interpreted as anything more than guidelines to aperture performance. As a general caution, Williams & Coletta (1987) comment:
"The concept of an optical resolution limit has proven difficult to formulate; it is not clear that the resolving power of an optical instrument is a particularly meaningful figure of merit. Even the cutoff frequency of a diffraction limited system can be surpassed under some circumstances. The concept of visual resolution suffers from a similar lack of generality, and the theoretical limits of performance are set by the task as much as by the underlying visual architecture. The sampling theorem [that predicts resolution based on Nyquist foveal cone spacing] correctly specifies the highest frequency possible for image reconstruction without aliasing. However, it does not necessarily prevent an observer from extracting enough critical features of a supra-Nyquist grating to be confident that he sees it." ("Cone spacing and the visual resolution limit," 1987, p.1521)
We might expect, then, to find that published resolution limits are inconsistent with each other and have limited generality: and this is what we find.
John William Strutt, Lord Rayleigh (pronounced RAY-lee) was originally concerned with the linear diffraction artifacts produced in spectral lines by the rectangular slit of a spectroscope, and he observed in 1879 that "a double [spectral] line cannot be fairly resolved unless its components subtend an angle exceeding that subtended by the wave-length of light at a distance equal to the horizontal aperture" (in other words, the factor k must be greater than 1.0). He proposed as a criterion that "the central line in the image of one [spectral line diffraction artifact] coincides with the first zero of brightness in the image of the other." This is given as k = 1.22 or Ro = 1.22·λ/Do radians in the table above. In telescopic images it places the central peak intensity of each Airy disk over the minimum of the first dark ring of the other (diagram, right), where the merged minimum brightness between the two stars is roughly 70% of the peak brightness of each Airy disk. Note that the Rayleigh criterion is commonly used in astronomy despite the fact that Rayleigh later (1880) determined by experimental tests that resolution with circular apertures was practicable at k = 1.09.
Jean-Bernard-Léon Foucault (c.1876), observing a scale of evenly spaced dark lines, found a resolution limit of about 4/3" with a 100 mm telescope, which is Ro =133"/Do and k = 1.17 at λ = 0.00055 mm.
William Dawes established his criterion in around 1832 after he "examined with a great variety of apertures a vast number of double stars, whose distances seemed to be well-determined," then calculated that a one inch (25.4 mm) aperture would just resolve a matched pair of 6th magnitude stars separated by 4.56 arcseconds (k = 1.02). He then extrapolated the ratio 4.56"/Dinches to apertures as large as 30 inches (762 mm), with the bald assertion that "the separability of all magnitudes is nearly the same, provided the state of the air is such as to bear well the increase of power."
C.M. Sparrow proposed (1916) that two point images can be resolved when the joint intensity function (the luminance sum of the overlapping diffraction artifacts) falls below a straight line connecting the two intensity peaks (k = 0.95). Note that no dark interval is perceived between the two point images, which does not qualify as resolution in some uses of the word. In two stars of equal magnitude, this means the intensity function between the two peaks of a point source pair is not perfectly flat but very slightly concave, and the visible circumferences of the Airy disks appear to touch (diagram, above). This is considered a more realistic resolution limit for both photographic optics and human visual acuity.
Other criteria exist: the Schuster criterion requires the full width Airy disks to be completely separated, which is twice the Rayleigh criterion; the Houston criterion uses the FWHM in the same way. These are rarely used in astronomy.
Because these criteria reflect assumptions that are questionable in some circumstances and arbitrary in others, I feel there is clarity in the choice of Ro = λ/Do (that is, k = 1.00). This lambda-D resolution criterion directly implements the 1/D rule first deduced by Fraunhofer (1823) and verified by the quantum theory of light. Empirically, it is equivalent to the average separation between two adjacent dark interference bands in the diffraction artifact of a circular aperture (the smallest diffraction detail, average k = 1.01), and is the criterion often used for the resolution of line pairs (see formula , above). It is insignificantly different from both the empirical Dawes resolution criterion (k = 1.02) and the FWHM diameter of the Airy disk (k = 1.03) as shown in the diagram (above). Whether this λ/Do interval can be resolved by the observer's eye depends on the image magnification and the observer's visual acuity. At an exit pupil of 1.0 (where magnification is equal to the aperture diameter in millimeters) it requires a visual resolution limit of Rv = 113.4 arcseconds, which is easily within the range of normal vision.
However, magnification and visual acuity are not the only factors that affect a practical resolution criterion:
Sharp eyed and experienced observers are able to detect a "rodlike" or "elliptical" elongation of the star image when two stars are separated by as little as k = 0.37 to 0.50 (a resolution threshold of Rv = ~0.45), although this detection criterion seems to rapidly deteriorate in apertures greater than about 90 cm because large apertures are limited by the effects of atmospheric turbulence.
Sidgwick (Amateur Astronomer's Handbook, pp.45-7) notes that matched 9th magnitude stars (fainter than the 4th to 6th magnitude range in which resolution is typically evaluated) require a separation of from k = 1.37 to 1.90, which implies a low contrast or low luminance criterion of roughly Rv = 135 to 190 arcseconds.
All the resolution criteria place the Airy disk of a binary component within the diffraction rings of the primary; these obscure the fainter star if the primary star is bright (m < 5), produces glare or scatter in the image, or if the companion is faint m > ~7.5 and is more than 2 magnitudes fainter than the primary.
The more stringent Dawes resolution criterion applies only to two stars of equal brightness viewed through a circular telescope aperture with no central obstruction. In addition, the Sparrow criterion becomes larger as the magnitude difference between the two stars increases. The Rayleigh criterion is applied without regard to the magnitude or difference in magnitude between two stars, the presence of a central obstruction, or differences in aperture.
Reflectors, because of the central obstruction, push some light into the diffraction rings, slightly reducing the brightness and diameter of the Airy disk. This can provide improved resolution for very close pairs of similar magnitude, but can obscure a faint companion located within the diffraction rings of the brighter star.
The reference wavelength of light is usually chosen to match the peak foveal response, which is not well defined. Different measures of the photopic sensitivity function place the average peak response between 545 and 555 nm, and individual differences in color vision enlarge this over a rather broad range between approximately 530 to 580 nm. The choice of 550 nm is purely conventional.
Appearance of the Diffraction Artifact. Given a fixed aperture and focal ratio, optical theory stipulates that regardless of the magnitude of the "point source" the diffraction artifact will always have the same linear and angular diameter and the same proportional illuminances between the Airy disk and any of the rings around it.
Vasco Ronchi (1961) noted the important difference between the ethereal image or unseen physical characteristics of the actual star, the calculated image or theoretical representation of this image in an optical instrument, and the detected image that actually appears to the astronomer's eye. Remarkably, optical theory in the form of the calculated image, a "mere mathematical construction," does not describe the detailed visual appearance of a stellar detected image.
First, as Sidgwick explains (Amateur Astronomer's Handbook, p.39):
"At normally encountered /ratios (say, /5 to /20) no intensity gradient across the disc is perceptible, the border between the disc and the first minimum appears nearly sharp, and the rings are brighter than theory would indicate, the first ring being not much fainter than the disc itself. ... The visible extent of the disc, like the number of rings visible, varies for a given instrument with the brightness of the source, although the discs are in fact the same size, irrespective of brightness."
In the 17th and 18th centuries the variable diameter of the Airy disks led some astronomers to believe that they were observing the surfaces of stars of different diameters or distances. However early 19th century astronomers such as William Herschel concluded this was an error, and coined the term spurious disc to denote it. In 1828 John Herschel declared flatly that "these are not the real bodies of stars, which are infinitely too remote to be ever visible with any magnifiers we can apply, but spurious, or unreal images, resulting from optical causes." (Note that his intent was to disparage the disk as the image of a physical object, not to disparage the visible disk in contrast to its mathematical description, as some amateur astronomers use the term today.) He continued:
"The apparent size of the disc is different for different stars, being uniformly larger the brighter the star. This cannot be a mere illusion of judgment; because when two unequally bright stars are seen at once, as in the case of a close double star, so as to be directly compared, the inequality of their spurious diameters is striking; nor can it be owing to any real difference in the stars, as the intervention of a cloud, which reduces their brightness, reduces also their apparent discs till they become mere points." (Encyclopedia Metropolitana, 1828)
The paradox of the varying disk diameter and number of rings was explained by Airy in 1835 as due to the luminance threshold of the eye. This is illustrated by plotting the diffraction artifact using a log scale for the luminance (diagram, right), because the relative brightness of lights viewed against a dark background is a logarithmic function:
Δb = b2/b1 = 100.4(m2-m1)
The dotted horizontal lines in the diagram show the relative effect of a visual threshold that extends down to either ~1/10th to ~1/1000th the peak Airy disk brightness. If a magnitude 2 star is bright enough to present three rings, this explanation requires the third ring of a mag. 2 star to appear as bright as a magnitude 9.5 star (diagram below, A).
Surprisingly, a threshold explanation for the changing angular size of the disk does not logically explain the appearance. For example, the Rayleigh resolution criterion results in a clearly visible dark gap between the Airy disks of two equal magnitude stars, which the diagram of the joint intensity function (above) shows is a brightness of about 73% of the peak disk brightness. But the brightest (first) ring of the diffraction artifact is only 1% of the peak brightness (table, above), and therefore should always appear to be "dark" (invisible) for any matched pair that appear to be two separate disks at the Rayleigh angular separation.
In addition, a faint star in a large aperture reflector can still display the first diffraction ring even when the Airy disk is noticeably reduced in size, and the diagram (above) shows that this is also inconsistent with a threshold explanation the threshold that would produce a reduced disk will be far above the peak brightness of any ring, so the ring would be invisible.
Finally, I observe the ring to contract around this smaller disk, maintaining the same dark gap width but becoming slightly thinner (diagram right, B). The ring certainly does not keep the same diameter regardless of the star magnitude, as theory requires: this would produce in very faint stars a visibly enlarged first gap around a much reduced Airy disk (diagram right, C). This contraction of the first ring around the reduced Airy disk also cannot be explained by the threshold model.
Magnification is the increase in the angular size (apparent width) of an object in comparison to the naked eye size of the object. In telescope optics, magnification results from the combined effect of the telescope objective and eyepiece. But the magnification is obtained in opposite ways: by means of projection magnification in the objective (a function of o), and by means of dioptric magnification in the eyepiece (a function of 1/e). In both cases, image dimensions formed by the naked eye are the benchmark to calculate magnification.
The Near Point. The eye has a variable focal length: it can focus on objects at different distances within the fixed distance from the cornea and lens to the retina. An object appears larger is effectively magnified as it is moved nearer to the eye, and this also increases the distance behind the eye at which the object image comes to a focus. To compensate, the eye lens can increase its optical power or accommodate to shorten the back focal length and bring the image of near objects into focus on the retina.
What object distance provides the benchmark for the calculation of magnification? It is the distance that provides the maximum image size on the retina, the minimum distance at which an object will still appear comfortably and completely in focus. This is the point of maximum accommodation, where the lens has contracted to produce its greatest optical power (shortest focal length).
Dioptric (Eyepiece) Magnification. An object moved closer to the eye than the near point will appear larger, but also out of focus because the lens cannot increase its optical power any further. Focusing the image of this closer object is made possible by adding the refractive or dioptric power of a magnifier lens or eyepiece.
Dioptric power is measured in diopters, the reciprocal of focal length (1/) measured in meters. The inverse form indicates that a shorter focal length means greater refractive power.
Magnification can be defined as the ratio between the half angular height β of the object at its magnified (apparently closer) distance to the eye divided by its half angular height α when viewed by the naked eye at the near point distance of 25 cm, and using the tangent of the angles for precision (diagram, above):
 Me = tan(β)/tan(α)
The distance size rule states that the ratio between the tangents of the two visual angles is the inverse of the ratio between the two distances. Thus, an object with an angular height α = 4° that is magnified to an angular height β = 20° will appear magnified by Me tan(20)/tan(4) = 5.2 times, and therefore will appear enlarged as if placed at a distance 250/5.2 = 48 mm (4.8 cm) from the eye.
This principle leads naturally to the calculation of the eyepiece magnification as the focal length of the eyepiece standardized on the near point distance of 250 mm (diagram below, right):
 Me = 250/e
Projection (Objective) Magnification. The projection magnification of a telescope objective is fundamentally different from eyepiece magnification: it is determined by focal length (o), not by the convergence angle of optical power.
The comparison here is to the magnification produced by a pinhole camera, which is produced only by the distance between the pinhole and the surface receiving the image. A pinhole has no optical power (no focal length), so to determine projection magnification we use the angular separation α of two rays from an imaged object.
In optics, a chief ray is the light path on which a lens or mirror has no optical effect, because it leaves an optical element at the same angle that it entered. This ray projects a greater field height y, measured perpendicular to the optical axis, as focal length increases. Therefore the magnification produced by the telescope objective is a linear function of its focal length (diagram above, left).
What telescope focal length will produce an image of a distant object (such as the Moon) that, when the image is viewed at the near point, will appear exactly the same size as the naked eye view of the Moon in the sky (x)? The answer: when o = 250 mm. So the objective magnification is equal to:
 Mo = o/250
For example, the objective in a 254 mm /8 telescope has a magnification of (2032)/250 = 8.13 times.
The Moon is 3475 km in diameter at a distance of 384,400 km, but in terms of optical calculations the linear width of its image at the near point is 2.26 mm. Therefore its image diameter at the focal plane of an o = 2032 system is 8.13 x 2.26 = 18.4 mm.
Telescope Magnification. A telescopic image is actually the compound of two different kinds of magnification: projection magnification that increases with increasing objective focal length, and dioptric magnification that increases with decreasing eyepiece focal length.
Because one magnification has a multiplying effect on the other, the magnification of the combined telescope and eyepiece system is the product of the separate telescope and eyepiece magnifications. The value 250 cancels out, leaving the ratio of the two focal lengths:
 Mt = Mo·Me = o/250 x 250/e = o/e
Or, starting with formula , and defining the tangents as α = h/o and β = h/e, we have:
Mt = β/α = (h/e)/(h/o) = (h/e)·(o/h) = o/e
Thus, in a 10" (254 mm) /8 telescope with a 20 mm eyepiece, the magnification is:
Mt = (254 x 8)/250 x 250/20 = 8.13 x 12.5 = 101.6
The diagram (below) summarizes these relationships. The magnification of the objective Mo depends on the distance that the real image is projected beyond the objective near point. The magnification of the eyepiece Me depends on the distance that the virtual image is located in front of the eye near point (green arrows in diagram). The optical effects are reciprocal: objective magnification, as in 17th century instruments, made telescopes larger (with larger relative apertures); high power, wide field dioptric magnification, the main innovation of 20th century eyepieces, allowed telescopes to be more compact (with smaller relative apertures).
Note that the field stop linear radius h is transformed into the virtual field stop limiting the apparent field of view (the eyepiece projection angle β) by the eyepiece focal length; it can be calculated from those parameters without actually measuring the physical edge of the field stop in the eyepiece. Also, as a feature of the eyepiece, the field stop does not define the maximum image width that can be produced by the objective, only the maximum image width that can be viewed through the eyepiece. The largest feasible image diameter is usually the interior diameter of the largest eyepiece that can be used with the objective, which in commercial 2" eyepiece barrels is about 1.8 inches or 46 mm.
When calculating magnification, keep in mind that manufacturer supplied data on telescope focal length, eyepiece focal length and eyepiece field stop diameter is often inexact, and therefore magnifications calculated from them should be considered approximate. Dioptric magnification greater than about 10x (e < 25 mm) and free of serious aberrations is not possible with a single lens; a compound lens is required.
Magnification & Image Brightness. Magnification affects the image illuminance: as magnification increases, the image becomes dimmer. This is because magnification causes a smaller part of the sky (the true field of view) to be passed to the eye, and therefore a smaller quantity of light is being used to fill the apparent field of the eyepiece. The magnification required to produce an exit pupil of a certain size, and the effect of this on image brightness (illuminance), can be calculated starting with the exit pupil equality de = Do/Mt = δ and assuming an eye pupil nominal diameter of 6 mm (table, below).
Thus, an exit pupil of 2.0 transmits only (2/6)2 = 1/9 or 0.1111 the light of an exit pupil of 6.0, which is produced by a magnification of 100x in a 200 mm objective or 175x in a 350 mm objective.
The telescopic image of an extended object (the Moon) cannot be brighter than the naked eye image of the same celestial area, due to "stopping down" of larger beams by the eye pupil. The telescopic image of a star brightens with aperture, because the larger quantity of light gathered by the larger aperture is concentrated into a star image that, at exit pupils above ~2.0, effectively remains a "point" source of light.
Various guidelines have been proposed to decide the optimal or most useful astronomical magnification. In this nook of the astronomical literature there are three strategies or approaches, based on exit pupil, target angular width, or image quality.
The first two approaches require the observer's visual resolution limit (Rv), the minimum angular width that can be resolved by the naked eye. Estimates of this limit for a normal eye range from 60 up to 300 arcseconds, depending on the type of resolution required. Typically 120 arcseconds is the value used for visual double star astronomy, although the limit becomes larger for faint, low contrast (deep sky) image detail. (Your personal resolution value is easy to measure with an aperture mask and a zoom eyepiece or a series of eyepieces of different focal lengths.)
Three Magnification Criteria. Aperture (Do) determines the dimensions of the diffraction effects that limit the angular resolution of the telescopic image. In some situations it is useful to select magnification in relation to the visualization of diffraction artifacts or resolution limits. This is the exit pupil approach.
The effects of a smaller exit pupil include: (1) greater visibility of the diffraction artifacts in the image, including the resolution limit of the aperture; (2) greater visibility of the effects on diffraction limited resolution of objective aberrations such as lateral color, coma or astigmatism, (3) a reduction in the effect of aberrations created by the observer's eye, due to concentration of the image beam at the center of the lens and cornea, and (4) the shadow projection of distracting "floaters" in the eye or of dust particles on the eyepiece eye lens.
If Rv is known (or assumed), then the optimal magnification can be determined by the angular size of the diffraction resolution criterion (Ro) of the aperture as:
Mopt = k·Rv·(Do/113) = k·Rv/Ro
where Do is measured in millimeters, 113 signifies the λ/D resolution criterion, and k is a "comfort factor" that depends on how much larger than this "just barely" visual limit you want the feature to appear. For example, the Rayleigh resolution criterion uses k=1.22, and in a 140 mm /7 telescope this is 1.22·113/140 = 1 arcsecond, which (assuming Rv = 120) requires a magnification of 120x obtained with an 8 mm eyepiece.
This magnification is sufficient to make all the image detail just visible to the observer's eye. As an analogy, it is equivalent to holding a printed image just close enough to the eye that the halftone dots in the image are visible, since the dots are the smallest detail the image contains. Any lower magnification will disguise the smallest details, and any higher magnification (for example, k = 2) will make the diffraction details larger and easier to see.
Because magnification by definition makes the image of the object you observe larger, it can be recommended by considering the angular size ρ of the smallest feature you are attempting to see, calculated as:
Mopt = k·Rv/ρ
Here k = 2 to 3 is more common. For example, a typical crater on the Moon is 10 kilometers wide, or about 5 arcseconds when viewed from Earth, which in a 140 mm /7 telescope requires a comfortable magnification (assuming k = 3) of about 3·120/5 = 70x obtained with a 14 mm eyepiece. This object magnification approach guarantees that the object you are looking for will be visually small but nevertheless clearly visible.
However, magnification influences many other aspects of the image besides the visibility of diffraction artifacts or the apparent size of features in object space. These include: (1) a darkened sky background that increases the visibility of faint stars or the contrast in extended faint objects such as nebulae, (2) a reduction in objective optical aberrations like coma, distortion or field curvature that increase with field height on the image plane, (3) a fainter and less distinct image, (4) amplification of the visual effects of atmospheric turbulence, focus inaccuracy and instrument vibrations, and (5) greater difficulty centering and manually tracking a target object. A suggested magnification based on any of these purely visual criteria has adopted the image quality approach.
Image quality is a vague criterion, because it is judgmental rather than calculated, and because image quality depends on many things, for example the optical quality of your objective, the angular width of the object or feature you want to see, collimation, light pollution, and so on. The recommendation to limit your magnification to "what the seeing will allow" is one example of the image quality approach: there is no consideration of the exit pupil or object apparent size involved in this suggestion.
Magnification Recommendations. Whether it is useful to see the resolution limit of the aperture or the smallest feature of an astronomical object depends on whether the image structure or the object structure is the observational interest. Whether you are able to use the magnification necessary to see the artifact or feature you are looking for depends on the quality of the image that results. This in turn depends on your equipment.
As a practical reality, the available configurations of amateur astronomical equipment are limited. The available range of apertures is approximately 80 mm to 600 mm; relative apertures range from /4 to /20, eyepiece focal lengths vary from about 5 mm to 40 mm. By combining these extremes we can estimate the equipment limit values for exit pupil (0.25 to 10), magnification (8x to 2400x) and resolution (1.41" to 0.19") that suggest practicable opportunities. Suggested magnifications are choices within those ranges supported by good visual reasons.
Magnification that is defined as a multiple of aperture (in inches or millimeters) is actually a recommendation for the diameter of the exit pupil (de), because the exit pupil can be defined as aperture divided by magnification (Do/M; see below), so magnification is just aperture multiplied by the reciprocal of the exit pupil, Mt = 1/de·Do. Magnification that is defined per se, for example as 200x, is specifically an object magnification recommendation, and is usually given in relation to a specific astronomical target (e.g., the Cassini division in Saturn's rings, the disk of Venus, a 3" separation in a binary star, the Hyginus Rille on the Moon). Image quality recommendations can use either the exit pupil or object magnification.
The astronomical literature commonly adopts the exit pupil approach, both because magnification varies inversely with exit pupil and because absolute magnification values (such as 200x) can potentially be anywhere within the range of magnifications possible with a single telescope and eyepiece selection, meaning they can have widely different image quality implications.
The five common exit pupil recommendations, and the justifications offered for them, are:
Mt = Do/δ, e = δN Normal magnification This defines the magnification at which the telescope exit pupil equals the eye pupil aperture of the observer. The rationale for this limit is that any lower magnification produces an exit pupil that is larger than the eye pupil, so that not all the light can enter the observer's eye: this effectively stops down (reduces the light grasp) of the objective.
This is a dubious rule. Few observers have actually measured their dark adapted pupil aperture (the customary value of 7 or even 8 mm is typically too large ... mine is 5.9 mm), and observing bright objects such as the Moon can contract the pupil to 3 mm or less. Low magnification has benefits unrelated to light grasp: the true field of view is larger, pointing and tracking become easier and less critical, and the effects of poor seeing are minimized. Finally, commercially available eyepiece focal lengths rarely exceed e = 40 mm, so telescopes at relative apertures above /6.7 (most refractors and Newtonians, and all Cassegrain format reflectors) will not reach the prescribed "normal magnification" anyway. Dobsonian users are most often concerned (needlessly) with this lower limit. A more useful benchmark is:
Mt = 0.25Do to 0.33Do, e = 3N to 4N Minimum useful magnification This is determined by the angular diameter of the target (star cluster, extended nebula, etc.) or by the focal ratio of the telescope, which can introduce distracting aberrations in wide field eyepieces at low powers. A magnification between 0.25Do and 0.5Do is useful for deep sky observing when a turbulent atmosphere precludes detailed observation of planets or the Moon. Thus, this rule is not based on an optical calculation but depends on the type of telescope, the type of target being looked at, and the apparent field of the eyepiece.
Mt = 0.5Do, e = 2.0N Minimum optimal magnification This is the magnification that causes the smallest (diffraction limited) image detail to be just visible, given a very optimistic estimate for the observer's visual resolution (i.e., ~1 arcminute or 60 arcseconds, which gives M = 60/Ro). Obviously, the quality of the observer's eyes affect this rule significantly.
Mt = 1.0Do, e = 1.0N Maximum optimal magnification This is the magnification that causes the smallest (diffraction limited) image detail to be just visible, given a conservative criterion for the observer's visual resolution (i.e., ~2 arcminutes or 120 arcseconds, which gives M = 120"/Ro). It is also easiest to remember, since the magnification is just the telescope aperture (in millimeters) and the necessary eyepiece focal length is just the telescope focal ratio. Note that this is equivalent to "Whittaker's rule", usually stated as Mt = 25Do when aperture is in inches, because 25 is just the number of millimeters in an inch.
Here is where we enter the skillful rather than rule based application of magnification. In many situations it is useful to try several eyepieces within the range of eyepiece focal lengths prescribed by the minimum and maximum "optimal magnification" rules. This is because variations in magnification cause sky brightness to decrease and make very faint stars more visible, and cause changes in apparent contrast in faint extended objects, making subtle details more visible: both effects usually occur within the range 0.5D ≤ M ≤ 1.0D. These image enhancements are often beneficial despite the somewhat increased visibility of atmospheric turbulence, but mediocre seeing, atmospheric dispersion and light pollution mean that conditions will dictate the best magnification to use.
Mt = 2.0Do to 4.0Do, e = 0.5N to 0.25N Maximum useful magnification This is the magnification that causes the smallest (diffraction limited) image detail to be just visible given the most pessimistic estimate for the observer's visual resolution (~5 arcminutes, which gives M = 300/Ro). This rule is highly dependent on observing conditions and the observing target. Planetary and lunar observers will generally prefer a lower magnification, within "the maximum that the seeing will allow" given the low contrast of planetary detail. In contrast, because the Airy disks of bright stars tolerate high magnification very well, even under poor seeing and especially in refractor telescopes, double star observers will use magnification at the high end of this range.
In all cases, the maximum useful magnification has been exceeded when the image is too dark (due to the magnification of the image), too blurry (due to the quality of the optics, to atmospheric turbulence), or too difficult to view (due to "floaters" in the observer's eye, small eye relief, etc.).
Despite its inherent ambiguity, the image quality rule use the magnification that produces the most useful image remains the most useful and trustworthy. Atmospheric turbulence, the magnitude of stars or the surface brightness and surface contrast of extended objects, the sky brightness, the type and optical quality of the instrument, the quality of the observer's eyes and the specific visual task or pastime all contribute to the choice of the optimal and most useful magnification. No formulaic exit pupil or magnification value can replace critical visual scrutiny of the image.
The corollary rule is: change the eyepiece often. No matter what the visual task or observing target, observing the same target with different eyepieces will verify the best choice, and can reveal unexpected details in the peripheral field or in the target. Magnification guidelines should be used merely to understand the image effects produced by an eyepiece of any specific focal length never to limit the range of eyepieces used.
Physical Attributes. The exit pupil is fundamentally the naked eye image of the telescope entrance pupil (aperture stop) as it appears through the eyepiece (image, right). If the telescope is turned toward the daytime sky, the exit pupil as a virtual image appears as a small, bright disk hovering just above the eye lens of the eyepiece. Like the virtual image in a mirror, it appears to move as you change your vantage.
This virtual image is formed by the convergence of all real image rays focused by the eyepiece. At this point the cross section of the projected, afocal real image of the telescope is physically smallest (as shown here). In this fixed position, centered on and perpendicular to the optical axis, it is traditionally called the Ramsden disc in the United Kingdom, in honor of the 18th century scientific instrument maker who first described it.
All real image rays means that a physical baffle with the same internal diameter as the exit pupil could be placed around it without obstructing any of the light exiting the eyepiece. The observer's correctly positioned iris mimics a baffle or stop when the telescope is used for visual astronomy; in that context the intersection of the exit pupil with the optical axis is called the eye point.
Finally, the exit pupil is the point where the telescope afocal image is most concentrated and therefore brightest; it is used as the benchmark luminance in the calculation of changes in image brightness caused by magnification.
These three prosaic physical attributes of the exit pupil virtual image, smallest real image, brightest projected image belie the almost fetishistic importance that some astronomers attach to it. In the evaluation of an optical system, as Smith, Ceragioli & Berry (2012) recommend doing it, "all that matters is the diameter of that exit pupil!" That claim invites a detailed examination.
Definitions. A commonly suggested method to specify the exit pupil (which is always stated in millimeters) is to measure its diameter with a reticule magnifier. This sometimes precedes the use of the exit pupil in the calculation of system magnification as the ratio between objective aperture (Do) and exit pupil diameter (de) that is, as the beam compression of the system:
 Mt = Do/de
Measuring the size of the exit pupil is inconvenient (especially when it is small), so it is simpler and usually more accurate to calculate the telescope magnification as the ratio of objective focal length (o) to eyepiece focal length (e). Then the exit pupil diameter can be calculated as the ratio between the objective aperture and the telescope magnification:
 de = Do/Mt
The logic of this definition is rarely explained. It treats the exit pupil as the naked eye image of the aperture stop, but replaces the objective/eyepiece combination with a single objective lens whose magnification is equal to its focal length (o). As explained above, this single objective will have a magnification Mo = o/250. Substituting, we have
de = Do/Mo = Do/(o/250) = 250·(Do/o) = 250/No
So the exit pupil calculation  based on magnification actually derives the exit pupil as a ratio of the relative aperture (as 1/No) of an equivalent single objective system.
This single objective must have a very large relative aperture to achieve the necessary magnification for example, No = 250 when de = 1.0 mm. Because the eyepiece has been omitted, this single objective does not form a Ramsden disc of real image rays. We can, however, make an appropriate comparison. If we place a 1 mm exit pupil at the near point, 250 mm from the eye (the distance of "actual size" magnification), its angular width will be 1/250 radian (0.23° or 13.75 arcminutes), and this matches the angular width of the single objective entrance pupil when viewed from its /250 focal distance. The exit pupil is then identical to the image of the aperture.
The significance of relative aperture in the system magnification of formula  becomes obvious if we substitute the common formula for magnification mentioned above (Mt = o/e), which yields a second definition of the exit pupil:
 de = Do/(o/e) = e·(Do/o) = e/No
Again, start with a single /250 objective; as we have already discovered, this will have an aperture image equivalent to an exit pupil de = 1.0. The entrance pupil, viewed at the focal distance, will subtend an angular width of 1/250 radians = 0.23° or 13.75 arcminutes. But at smaller relative apertures the focal plane will be closer to the entrance pupil and therefore the aperture image will appear much larger: for example, in an /10 objective, the entrance pupil will appear to be 5.73° wide.
Eyepiece magnification, because its effect on image magnification is the same as extending the objective focal length, is equivalent to reducing the entrance pupil angular size by division. As explained above, eyepiece magnification is calculated as Me = 250/e. So an e = 10 mm eyepiece (which according to formula  yields an exit pupil of 1.0 in the /10 system) provides a magnification of 250/10 = 25x, and 5.73°/25 = 0.23° or 13.75 arcminutes. As before, the telescope exit pupil viewed from the near point is identical to an objective entrance pupil of equivalent magnification viewed from its focal plane.
In practice, the exit pupil can be changed in three ways: by changing the eyepiece focal length e, by using a Barlow lens or focal reducer that increases the effective relative aperture No, or by applying an aperture stop that increases the relative aperture No. The diagram (below) summarizes the relationships in the definition of the exit pupil between eyepiece and magnification, and between aperture and relative aperture.
Changing the objective focal length o changes the relative aperture and magnification directly and in the same direction, scaling the whole system up or down. In contrast, increasing the aperture Do or eyepiece focal length e changes the size of the system relative aperture No and magnification Mt in the opposite direction. As the numerator of one formula increases, the denominator of the other formula decreases, which maintains the equivalence between formulas  and .
Object vs. Image Magnification. These apparently opposing effects of aperture and eyepiece are related to the two contrasting ways to produce magnification: by (1) projection magnification, the increase in the apparent size of a distant object (the astronomical target) delivered by focal length alone acting as a "pinhole" system (see formula ); or by (2) dioptric magnification, the increase in the apparent size of a nearby object (the image plane) delivered by the refractive magnification of the eyepiece alone, acting as a magnifier (see formula ). For clarity I will refer to these with the nonstandard terms object magnification and image magnification, respectively.
What determines the desirable object or image magnification? The ability of the eye to discern some criterion interval of smallest width. The conventional method to define this "useful" magnification is in relation to an arbitrary interval k of the dimensions of a point source diffraction artifact. The diffraction artifact represents the resolution limit of the telescope, because at this angular width the image of a target object dissolves into the diffraction pattern of quantum light interference (described above) in any circular aperture. This connection to the telescope diffraction limit defines the importance of the exit pupil for many astronomers.
Because we have two definitions of enlargement to work with, we require two matching definitions of minimum diffraction width: as (1) an angular width (visual extent in image space) expressed in arcseconds, and as (2) a linear width (physical extent on the image plane) expressed in millimeters. Setting aside for the moment the arbitrary interval k, the underlying metric or unit of measurement in each case is calculated from a wavelength of light (λ) that defines the dimensions of diffraction:
[4a] angular width: Ro = 206265·λ arcseconds
[4b] linear width: So = 250·λ millimeters
where 206265 (the number of arcseconds in a radian) projects the metric into naked eye image space, and 250 (the relative aperture No of unit magnification) projects the metric onto a plane located at the near point. The metric defined by diffraction bands of "yellow green" light (λ = 0.00055) yields Ro = 113.4 arcseconds of angular width and So = 0.1375 mm of linear width.
These intervals are abstract proportions in the realm of optics and physical wavelengths of light. To determine whether these intervals will be visible to the human eye, they must be compared to a visual resolution limit defined in the same way. So we need to define the smallest interval of visual resolution as (1) a minimum angular separation in cycles per degree or equivalent arcseconds (Rv), and (2) a minimum linear spacing in a square wave grating viewed at the near point distance (250 mm) and stated as line pairs per millimeter or the reciprocal single pair width in millimeters (Sv). Often cited values for a single line pair viewed by a normal adult eye under adequate illumination are:
angular width: Rv = 120 arcseconds
linear width: Sv = 0.1454 millimeters.
Keep in mind that these are population averages and, depending on the condition of your eyes, your individual visual acuity may be larger or smaller than the normative values. For example, my naked eye resolution of a telescope image is approximately Rv = 100 arcseconds.
If the telescope and visual resolution limits are defined with the same angular or linear metric, we can define the exit pupil at which visual resolution of diffraction limited detail will occur as the ratio of the two:
 de = Ro/Rv = So/Sv
and when Ro = Rv then de = 1.0 by definition. That is the logical result of the fact that we choose an optical resolution criterion that matches the observer's naked eye visual capability. The fact that Ro is already close to Rv indicates that the normal eye is essentially diffraction limited.
 Mt = Do(Rv/Ro) or Do(Sv/So)
 e = No(Ro/Rv) or No(So/Sv)
Note that the ratio of optical to visual resolution is inverted between the two formulas one definition is the reciprocal of the other. (The more object magnification is provided, the less image magnification is necessary.)
All that remains is the choice of k, the arbitrary diffraction interval taken as the smallest (diffraction limited) angular or linear extent of image detail. For reasons explained above, I prefer k = 1.0, the λ/Do criterion, which directly adopts the metrics defined in [4a] and [4b] above (Ro = 113.4 arcseconds, So = 0.1375 mm) and is approximately the average "line pair" separation between two diffraction rings in a star diffraction artifact, the smallest observable diffraction detail for a given aperture.
If the Rayleigh resolution limit (k = 1.22) is desired instead, it is found by multiplication: 1.22*Ro = 1.22*113.4" = 138.4". The Dawes limit is 1.02*113.4" = 115.7"; the Sparrow limit is 107.8". (See the table above.) If the criterion interval is the Airy disk, with a FWHM diameter of k = 2.06, then 2.06*Ro = 2.06*113.4" = 233.6". (It is often asserted that the smallest diffraction detail is resolved when the exit pupil is 1.0, but as I've shown this only occurs when Ro = Rv or So = Sv.)
The following table illustrates these relationships using the λ/D resolution criterion at a wavelength of 0.00055 mm (550 nm), for (1) apertures Do with the angular resolution criteria Ro = Rv = 113.4 arcseconds, and (2) relative apertures No with the linear resolution criteria So = Sv = 0.1375 mm. These define the object magnification (Mt), eyepiece focal length (e) and image magnification (Me) necessary to resolve the criterion interval of the diffraction artifact.
This table makes clear that the exit pupil provides no information about object magnification in the absolute sense. In the table, there can be a constant exit pupil across all magnifications, or a complete range of exit pupils within any single magnification. If we point our telescope at the planet Jupiter, and we know that the telescope image is produced at an exit pupil of 1.0, we cannot conclude anything about how large the image of Jupiter will appear through the eyepiece. This is because we have been considering object magnification entirely as the product of a virtual focal length (t =Mt·250 mm), and focal length gives us no information about aperture.
Instead, exit pupil is inversely proportional to the image magnification necessary to resolve the diffraction artifact, calculated as the ratio of eyepiece focal length to objective relative aperture (formula , above). It is also inversely proportional to the object magnification necessary to resolve diffraction limited resolution, calculated as the ratio of aperture to magnification (formula , above).
In the table, there is a constant exit pupil across all relative apertures because we proportionately increase the eyepiece magnification: a smaller relative aperture No produces a smaller linear width in the diffraction artifact, and this smaller dimension requires a shorter eyepiece focal length (greater dioptric magnification) to resolve. Similarly, there is a constant exit pupil across all apertures, because larger apertures create a smaller resolution interval, which requires a longer focal length (greater object magnification) to resolve.
The dependence of diffraction magnification on relative aperture can be demonstrated with an aperture mask. If the eyepiece is not changed then the object magnification remains constant (because objective focal length also is not changed), which can be verified by observing through the mask the constant angular separation of a bright double star such as bet GEM (Castor) or bet MON. Despite this constant object magnification, the mask creates a smaller aperture, which greatly increases the relative aperture; this reduces the exit pupil and increases the apparent size of the diffraction artifact.
"Wasted" Light & "Empty" Magnification. Because the exit pupil is just the ratio between the objective and visual resolution factors, it is often used to select an eyepiece magnification suitable for the parameters of any specific telescope system. It's easy math to multiply focal ratio by the desired exit pupil value to get the necessary eyepiece focal length.
Handy when applied across many different telescope systems, the procedure is unnecessarily used to define the factitious twin "problems" of wasted light (or wasted aperture) and empty magnification.
The astronomer is often advised to prevent "wasted" light by avoiding magnification that creates an exit pupil larger than the dark adapted eye pupil, where de > δ. The argument is that an eye pupil smaller than the exit pupil will block some light from entering the eye, effectively stopping down the objective aperture.
However, with most commercially available astronomical equipment, it is unusual to exceed the eye pupil aperture. With an /6 telescope and a pupil aperture of 6 mm (my measured dark adapted pupil size), an eyepiece with focal length greater than 36 mm would be necessary. Even in an /4 telescope, only eyepieces above 24 mm would cause the problem. And this rule misses the point that a longer focal length eyepiece is useful for a wider, lower power view of an extended object. Low magnification is used to increase the true field of view at the cost of "wasted light", in the same way that high magnification is used to increase visual resolution at the cost of reduced image luminance. (The most serious drawback when de > δ is that image contrast is further reduced and fainter stars or extended objects become even more difficult to see.)
At the other extreme, "empty" magnification is said to be any enlargement greater than sufficient to make the smallest diffraction detail visible to the eye. The logic is that you are no longer resolving detail in the celestial object of interest (e.g., the surface details of a planet or the Moon) but resolving the diffraction that actually obscures detail. This limiting exit pupil can be calculated, for any visual resolution criterion Rv, as:
 de = 138.4"/Rv mm (Rayleigh criterion)
 de = 113.4"/Rv mm (λ/D criterion)
Thus, adopting the commonly cited visual resolution threshold of 120" and the Rayleigh resolution criterion predicts that the dark rings around the Airy disk will be resolved when the exit pupil is de = 1.15, and the Airy disk will be detectable when the exit pupil is twice as large (de = 2.3). Conventional wisdom asserts that an exit pupil half as large (de < 0.5 mm) only provides "empty magnification" because the eyepiece magnification is then more than twice the amount required to resolve the diffraction limited image detail.
This abstract reasoning ignores any situation where the diffraction artifact is actually what you want to observe. Because the Airy disk is quite robust to magnification when the seeing is mediocre, exit pupils below 0.2 are routinely valuable when resolving close, bright double stars and counteracting the image motion of turbulence. (William Herschel employed exit pupils as small as 0.03 when studying double stars with his "most excellent" /13 reflector.) Very small exit pupils are necessary when conducting a star test of optical quality. There is no such thing as "empty magnification" only magnification that is not optimal for the visual task at hand under actual observing conditions.
The linear diameter of the Ramsden disc does have important visual consequences. A small exit pupil (less than 1.0 mm) projects light like a pinhole, which can cause vivid shadowing of suspended internal debris or "floaters" on the retina. At the same time the tiny Ramsden disc must be centered in the plane of the iris aperture to produce a view of the entire eyepiece field, which becomes a kinesthetic challenge in very wide angle eyepieces or awkward viewing postures unless the observer has a chair or ladder for support. On the other hand, a smaller exit pupil is only refracted by the central portion of the eye lens and cornea, minimizing the effects of astigmatism. For that reason, many older viewers prefer smaller exit pupils.
More significant problems with the exit pupil occur when observing during daylight with a binocular or spotting telescope. The high luminance of daylight images contracts the pupil aperture to 2 mm or less, and this makes alignment of the eye pupil and exit pupil(s) more difficult. It also makes any spherical aberration of the exit pupil more noticeable.
Image Brightness. In addition to estimates of magnification in relation to diffraction limited resolution, the exit pupil is also used as a benchmark for relative image brightness. Here eye pupil diameter δ (also measured in millimeters) is required for the calculations.
The first  definition of the exit pupil represents the same proportion that defines the telescope image illuminance in relation to the retinal illuminance the ratio in the brightness of the telescope image to the brightness of the naked eye image of the same area of sky:
 It = (Do/Mtδ)2, therefore It = (de/ δ)2
As explained above, the size of the exit pupil is related to the system magnification, so that as magnification increases the true field of view gets smaller and the exit pupil, through which all the light exiting the telescope must pass, also gets smaller. As a result, variations in the exit pupil diameter affect the exiting illuminance of the telescope image in the same way that the opening in a camera diaphragm determines the amount of light that can enter a camera. As the exit pupil gets larger, the image luminance increases (graph, left).
The "shutter" and magnification effects combine to brighten or darken the image in inverse square proportion, and the difference in the illuminance of the telescope image equals the ratio of two exit pupils squared. The peak image brightness occurs when de = δ. At that point, the telescopic image is theoretically just as bright as the naked eye image of the same area of sky, omitting transmission loss in the instrument (which in practice is usually significant).
Referring again to the concept of the exit pupil as the image of a single objective entrance pupil viewed from the objective focal distance, an exit pupil of 6.0 indicates the image brightness of an /(250/6) = /42 optical system with an entrance pupil apparent diameter of 83 arcminutes (1.38°).
Because any larger exit pupil diameter will exceed the nominal eye pupil aperture, de > δ, the fixed eye pupil becomes the limiting factor as the exit pupil is increased. The area sampled by the fixed eye pupil size is a smaller part of the total exit pupil area, effectively stopping down the objective aperture at the output end. Image brightness again declines.
Apparent Field of View
The AFOV is measured as the projection angle of the radius (β) or diameter (2β) of the field stop (diagram, below). The tangent of this angle is equal to the radius of the field stop opening divided by the eyepiece back focal length (BFL).
The physical diameter of the eyepiece barrel and location of the field stop are dictated by the optics of the eyepiece (its abaxial aberrations and design AFOV) and the ratio of the objective (see here). The field stop excludes abaxial light from the objective that would produce aberrations in the eyepiece image or an unacceptably dark image, and blocks stray light reflected from the interior of the telescope tube. The barrel diameter limits the maximum internal diameter of the field stop, to ~27 mm in a 1.25" eyepiece barrel and ~46 mm in a standard 2" eyepiece barrel. In all well made eyepieces, the field stop will define the apparent field of view and will be crisply in focus.
The apparent field of view of an eyepiece is normally specified by the manufacturer, along with the eyepiece focal length and design type. However, the nominal AFOV of an eyepiece is often inflated due to the effects of positive distortion, especially in "wide field" eyepieces (AFOV > 60°).
When the manufacturer's apparent field specification appears inaccurate, or is not known, then the AFOV can be measured in three ways:
(1) Projective Measurement. Support the eyepiece so that it is lying on its side with the eye lens about 2 feet from a smooth wall or flat surface. (The seam between pages of an open hardcover book, or between two books of the same thickness laid spine to spine, is convenient.) Support a high powered flashlight in a similar way so that it shines into the field lens (barrel end of the eyepiece) from a distance of a few feet. (The eyepiece and flashlight should be lying perfectly level, the optical axis of the eyepiece and flashlight must be perpendicular to the wall, and the flashlight must be "collimated" or centered on and parallel to the eyepiece optical axis, which can be arranged by centering the shadow of the flashlight bulb inside the projected circle of light.) The eyepiece will project the light from the flashlight onto the wall as a circle of light larger than the flashlight beam. Project the image onto a small white card held close to the eye lens, and move the card back and forth until the projected circle appears smallest: this is the location of the exit pupil. Measure the greatest width of the circle (D) and the projection distance (P) from the eyepiece exit pupil to the wall. Then:
AFOV = 2arctan(D/2P).
For example, if the imaged diameter is 82.3 cm, and the wall to exit pupil distance is 73.3 cm, then:
AFOV = 2arctan(D/2P) = 2 x arctan(82.3/(2 x 73.3)) = 2 x 29.3° = 58.6°.
(2) Visual Measurement. Tape to a convenient wall a long strip of paper or tape measure on which small increments (inches or centimeters) are clearly marked and visible from a distance (Dm) of about 100 cm. Hold the eyepiece over one eye so that the full field is clearly visible and superimposed over the measurement tape, visible with the other eye. Manipulate the eyepiece position until the tape appears to measure the full diameter of the eyepiece field (De). Then:
AFOV = 2arctan(0.5De/Dm).
For example, viewing a metric measuring tape from 100 cm, you measure an apparent width of the eyepiece field as 93 cm. Then:
AFOV = 2arctan(0.5 x 93/100) = 2arctan(0.47) = 2 x 25° = 50°
(3) Physical Measurement. Use calipers to measure the interior diameter of the field stop, assuming it exists and can be recognized. If in doubt, look through the eyepiece and use the tip of a toothpick or pencil to locate the field stop: this is either the end of the eyepiece barrel or the interior edge of the lock nut holding the field lens. (Note that many "super wide field" eyepieces locate the field stop inside the eyepiece where it cannot be measured.) Then the apparent field is simply the visual angle α that the field stop radius would subtend at 250 mm, divided by the apparent field radius β. The distance size equation then makes this equal to:
AFOV = 2arctan(0.5Dfs/e).
For example, if the field stop diameter is 23 mm and the eyepiece focal length is 20 mm, then:
True Field of View
The exit window admits to the eye an image that has been magnified by both the telescope and eyepiece. The effective size of the exit window, if the image area it defines were viewed with the naked eye, would appear much smaller and much farther away (diagram, above). The area of sky that would be visible through that opening at that distance is equivalent to the true field of view (TFOV) of the objective/eyepiece combination.
Standard Formulas. The TFOV is equal to twice the maximum field angle (α) focused by the objective on the image plane that is admitted to the eyepiece by the physical radius of the field stop (h'). Since α is a linear function of the focal length (o), the TFOV is:
TFOVo = 2αmax = 2h'max/o.
By substitution, the TFOV is calculated as the apparent field of view (AFOV) divided by the image magnification (Mt):
TFOV = AFOV/Mt.
Thus, if the apparent field of view of an eyepiece is 50°, and the magnification of the telescope and eyepiece combination is 200x, then the true field of view is:
TFOV = 50°/200 = 0.25°; or 0.25° x 60 = 15 arcminutes
the multiplication by 60 converts the fractional degrees to arcminutes. In the diagram (above), an eyepiece with a 50° field of view offers the same visual area as a circular opening 23 cm in diameter held perpendicular to the eye at a 25 cm distance. Then the true field of view is equivalent to the area of sky seen through the same circular opening placed at a distance of 50 meters (164 feet).
If the field stop diameter and objective focal length are known, then the TFOV can be calculated as:
TFOV = 57.3·(Dfs/o) degrees.
This formula can also be used to calculate the maximum true field of view possible with an eyepiece barrel or a visual back of a certain diameter. For example, in a 254 mm (10") /8 telescope and a 1.25" (27 mm) eyepiece barrel, the maximum true field of view is:
TFOV = 27/2032 x 57.3° = 0.76° x 60 = 46 arcminutes.
Using a 2" (46 mm) eyepiece barrel, the maximum true field of view is:
TFOV = 46/2032 x 57.3° = 1.30° x 60 = 78 arcminutes.
Star Drift Measurement. If the field stop diameter and apparent field of view are not known, the true field of view can be measured by means of star transit times. This requires at least three measurements of the time (in seconds) it takes a bright star, placed outside the eyepiece field of view, to appear at one side of the eyepiece field, cross the center of the field, and disappear at the opposite side.
Care must be taken to hold the head in a fixed position, so that the entire circumference of the field edge is visible, and to pass the star through the center of the eyepiece field.
Then the true field is the average of the three timings multiplied by the cosine of the star's declination (stars closer to the celestial pole will require a longer time to transit the eyepiece field):
TFOV = 0.25cos(Declstar)·t.
The factor 0.25 is necessary to convert from seconds in time to arcminutes of angular width. For example, using Regulus (alp LEO) as the transit star (Declination = +11°58'), three eyepiece transit times are 134, 131 and 138 seconds. Then:
TFOV = 0.25 x cos(11.97°) x (134+131+138)/3 = 0.25 x 0.978 x 134.3 = 32.8 arcminutes.
Given the TFOV (in arcminutes) and system magnification, the AFOV (in degrees) is:
AFOV = TFOV / 60Mt.
The TFOV determined by star transit measurements will typically not correspond exactly to the TFOV calculated from the manufacturer supplied eyepiece apparent field of view and the magnification calculated from the manufacturer supplied eyepiece focal length. This is usually because the eyepiece AFOV and e are nominal, inferred from the computer optical design rather than the physical construction of the eyepiece and because star transit times includes some systematic measurement error due to curvature of field; taking transit times in a commercial SCT, whose focal length changes with mirror focusing, also can create small discrepancies.
These errors are typically not large. The chart (right) compares the TFOV of over 60 different eyepieces, as calculated from the nominal eyepiece AFOV and e and as measured by star transits. The calculated AFOV overestimates the measured TFOV by about 4% on average; the overestimation is not correlated with AFOV but is largest (up to 22% greater) at the shortest eyepiece focal lengths (highest magnifications) and in certain eyepiece brands or designs (Orthoscopics).
Richest Field Telescopes
A richest field telescope (RFT) is a telescope design and eyepiece selection that provides the brightest image within the widest practicable field of view or, in the original definition, that can show "more stars in the average Milky Way view than any smaller or larger telescope can show." This design criterion is dependent on three parameters: exit pupil, objective aperture and relative aperture (focal ratio), selected as follows:
Exit pupil (de) determines the diameter of the compressed light beam that is passed to the eye by the aperture. To ensure the brightest possible image, the exit pupil of an RFT must match the diameter of the observer's dark adapted eye pupil (nominally 7 mm, but in older observers typically 5 mm to 6 mm).
Aperture (Do) determines the limit magnitude of the system and therefore the total number of stars (or features of dim extended objects) that will be visible above the observer's dark adapted luminance threshold. Because exit pupil is fixed, the choice of aperture also defines the system magnification (Mt) and, by the selection of eyepiece apparent field of view, the telescope true field of view.
Finally, relative aperture (No) determines eyepiece focal length (e) as a multiple of the exit pupil (e = de·No). Since exit pupil is already fixed, the RFT principle limits the largest relative aperture to the practical upper limit of eyepiece focal lengths, including focal lengths that do not introduce astigmatism or spherical aberration of the exit pupil at low powers; and limits the smallest relative aperture to focal ratios that do not introduce visible coma or chromatic aberrations from the objective.
The limiting factors are the exit pupil and optical aberrations. Increasing aperture gathers more light, producing a brighter image but requiring the choice of an eyepiece with a longer focal length and typically more noticeable aberrations. Increasing the aperture causes the magnification to increase as well, reducing the true field of view and requiring the choice of a smaller relative aperture, again increasing aberrations. What is the optimal balance? To answer that question some measure of the system performance is required.
S.L. Walkden, in the original RFT paper (1936), proposed using an estimate of the number of stars per square degree of sky visible with a given aperture. This star count density is calculated as the magnitude limit of the aperture divided by the squared magnification that is delivered by the maximum exit pupil at that aperture. Given a specific aperture Do and exit pupil (de) measured in millimeters, the objective magnitude limit mD, aperture star count per unit area of sky (KD) and magnification adjusted star count RFTD can be estimated as follows:
mD = 2.8+5*log(Do)
KD = 10(0.0003*mD3+0.0019mD2+0.484mD3.82)
Mt = Do/de
RFTD = KD/Mt2
With this metric one finds that the peak star count is at around D = 70 mm (diagram above, blue curve); this is also Walkden's original conclusion, and illustrates his intent to find a unique advantage for small apertures in an era of giant telescopes. Other authors, for example Glenn Shaw (Sky & Telescope March, 1980), use empirical star counts in different parts of the sky, which give widely ranging results, including apertures at and above 200 mm. This demonstrates that a star count metric by itself is unreliable. Star count is a heuristic that leads to the concept of an RFT, but does not usefully define the parameters of an RFT.
However, scaling the RFT aperture on a rudimentary luminance weighted star count formula
RFTD = mD·KD/Mt2
produces a gently peaking curve at 150 mm with a broad maximum between 90 mm to 260 mm (~3.5" to ~10"; diagram above, yellow curve); these apertures deliver a large true field of view between 6.2° to 2.2° with modern super wide angle (80° AFOV) eyepieces. This again is consistent with Walkden's admission that a 7" aperture might be more suitable as an RFT, because "with the larger telescopes, the proportion of strikingly bright looking stars increases rather rapidly and the proportion of faint looking ones decreases rather rapidly; so that except in actual number of stars to be counted, the fields of the larger telescopes generally look richer, and decidedly more splendid."
These sentiments match the recommendation by Robert Ayers that the optimal RFT refractor has an aperture of 80 mm to 200 mm with a focal ratio between /4 to /6. There is a trivial change in the luminance weighted star counts across this aperture range, which verifies Shaw's observation that he could see no difference between 6" and 8" RFT scopes. And as a historical note, E.E. Barnard used a 250 mm, /5 Petzval or "portrait" lens for his high resolution, wide field and splendidly rich photographs of the Milky Way.
Given that the longest focal length offered in most commercial eyepiece lines is 40 mm (which gives a ~7 mm exit pupil at /6), and that optical aberrations become severe at relative apertures smaller than /4, we can restate the abstract RFT specification as follows:
an RFT is any fast focal ratio (No = 4 to 6) telescope used for visual astronomy; it must have a sufficiently small aperture (Do <260 mm) to deliver a 6 mm to 7 mm exit pupil when used with a low power (e = 24 mm to 42 mm), wide angle or super wide angle (AFOV >70°) eyepiece.
Ayers offers cogent arguments in favor of choosing a refractor over a reflector for an RFT although giant binoculars (D > 80 mm) may also be suitable for very wide fields. If one is interested to use the fastest feasible /ratio, a large aperture (~250 mm) coma corrected Newtonian reflector is an attractive alternative disregarding the vignetting effect of the central obstruction, which is rarely noticeable to the visual astronomer.
Effect of Central Obstruction
Resolution is affected by wave diffraction effects, but so too is the potential of the objective to transmit contrast information at different spatial frequencies. Transmission is specifically affected by a central obstruction, such as the secondary mirror in a Newtonian or Cassegrain design, which creates an annular rather than circular aperture geometry and diffraction at four edges on the diameter rather than just two. This obstruction ratio is defined as:
obstruction ratio (η) = Ds / Do
where Ds is the diameter of the secondary obstruction, the mounting for the secondary mirror.
The standard method to evaluate optical performance across a range of spatial frequencies is the modulation transfer function or MTF. The MTF shows the proportion of luminance contrast that is retained in the image formed by an optical system as the spatial frequency of the luminance variations is increased. Geometrically, the MTF calculation is equivalent to the area of overlap between two aperture superimposed at a displacement equal to the spatial frequency.
The stimulus used is either a pattern of crisp black and white lines (to evaluate the transmission of sharp edges) or a sine wave pattern of modulated grays (to evaluate the transmission of tonal gradients). Both methods converge on similar conclusions.
Modulation transfer is defined as the reduction in image contrast between the range of the luminance in the image (LI) as a proportion of the range of luminance in the target or stimulus (LS):
contrast ratio = (LImax LImin) / (LSmax LSmin)
Because diffraction produces a tiny amount of smearing in the image, contrast is imperceptibly reduced at low spatial frequencies (wide spacing) but becomes severe at high spatial frequencies (very narrow spacing). So the MTF is scaled to zoom in on the high spatial frequencies, defined as the inverse of the angular spacing in multiples of the minimum theoretically resolvable angular spacing (λ/Dmm). For example, the Rayleigh criterion of resolution is defined as 1.22λ/D, so its spatial frequency in an MTF is 1/1.22, or 0.82. (The MTF is sometimes graphed in terms of line pairs per millimeter (lp/mm) for a specific telescope relative aperture, but this produces the same focus on high frequency spacing and the compression of low frequency spacing at the far left of the graph.)
The green line in the generic MTF diagram (right) shows the curve for an optically perfect and unobstructed circular aperture transmitting a sine wave pattern with a spacing frequency up to the resolution limit which in a 250 mm /10 telescope is a stimulus angular separation of 0.45 arcseconds or a spacing on the image plane of 0.0055 mm or 182 line pairs per millimeter.
The yellow line shows the effect on the contrast ratio of a 25% obstruction, typically the upper limit in an /8 Newtonian reflector, and a 50% obstruction, the upper limit in a Cassegrain design optimized for astrophotography (Ritchey Chrétien or Schmidt Cassegrain). Both patterns can be summarized (reading the graph left to right) as (1) a negligible (less than 10%) contrast reduction in spatial frequencies from the full image diameter down to about 10 times the Rayleigh limit, (2) a significant drop in resolution down to about 2 or 3 times the Rayleigh limit, (3) an increase in resolution that can exceed an unobstructed aperture by a small amount (due to enhanced contrast caused by overlapping wave reinforcement), and (4) resolution matching the theoretical minimum at the Rayleigh limit.
Using the 50% obstruction curve as an example, the greatest relative loss of contrast occurs between 2 and 3 times the minimum resolution, or an angular width of about 1.5 arcseconds. This means the most severe image degradation will occur in lunar features that are about 3 kilometers wide, but will produce a slight crispening in features half that size. It will most affect features that are about 3% of the width of Jupiter's disk at opposition or about 6% the width of Mars's disk at opposition. Contrast on the Galilean satellites of Jupiter and on the disks of Mercury, Uranus and Neptune would be strongly obscured or obliterated. Double stars separated by about 1.3 arcseconds would be most compromised, but very close pairs would be easier to separate.
Whether and how much a central obstruction matters depends somewhat on the size of the aperture, but more on the specific task the telescope is used to perform. Astrophotographers can be tolerant of a large central obstruction because they typically work on an image scale of 1 pixel = 5x the resolution limit, and image contrast can be enhanced with image stacking and digital manipulation.
For visual astronomers the problem is more complex. Inherently high contrast targets such as the moon and double stars will display relatively minor contrast reduction, and the effect will depend on magnification, because human contrast sensitivity peaks at around 7 cycles per degree (around 10 arcminutes) and declines as frequency increases. At the other extreme, low contrast and dim targets such as deep sky objects will be minimally affected because the fovea cannot function at their low luminances. This leaves planetary observation and close double stars widely different in magnitude as the targets where the size of the central obstruction can be a critical issue, one reason that Newtonian or refractor telescopes are often preferred by planetary and double star observers.
Optics of the Eye
The eye is an equal component with objective and eyepiece in the performance of an optical system. The critical performance dimensions for astronomy are dark adaptation, hue discrimination, and resolution as a function of stimulus brightness and stimulus/background contrast. However I begin with a topic that is customarily minimized or ignored in discussions of visual performance: individual differences.
Human vision is generally characterized in three ways: by describing the physical structure and theoretical optics of the eye; by calculating the optical performance of "ideal" or mathematical eyes; and by measuring individual performance in a variety of visual tasks.
Visual performance is measured using tasks of detection (the stimulus is declared to be visible or not visible), matching (two stimuli are declared to be or are adjusted by the observer until they appear to be the same), or discrimination (similar stimuli can be identified as different in some way). In all cases, statistical theory is required to summarize performance across individuals. Stimulus thresholds in detection or discrimination are defined as the stimulus value that produces a 50% or 95% probability that an average person will respond with the necessary detection or discrimination. Matching stimuli are defined by the average and standard error of stimulus values judged to be the same, or the average just noticeable difference (JND) that produces detection that they are different. These measurements are then used to create models of average human performance, which deviate from actual performance.
Two examples will illustrate the scale of the problems. On the subject of star color, the first diagram (below left) shows the location on a standard hue circle of the range (variation) in the single wavelengths chosen as a "pure" or best shade of red, yellow, green and blue light by subjects with normal color vision; the inset shows the number of individual subjects (circles) who chose a specific wavelength in the "green" matching task. On the subject of calculated telescope resolution criteria, the second diagram (below right) shows the variation in the judged brightness of wavelengths across the visible spectrum; the "peak" wavelength, which is used to define aperture resolution criteria, varies across a 40 nm range from about 530 to 570 nm.
In the color matching task, there seems to be good agreement among subjects in the locus of a "pure" yellow wavelength in fact, this constitutes the Nagal anomaloscope test for red/green color blindness. But in the green and blue color matching the performance can best be described as scattered: the green values chosen by different subjects range from a "green blue" below 500 nm to a "green yellow" near 560 nm.
In terms of photopic luminance, individual peak sensitivity spans a range of at least 40 nm (from 535 to 575 nm). The commonly used value of 550 nm is near the peak of the 1964 CIE V* photopic luminosity function (orange curve) or a similar function, which are actually not defined by tests of brightness matching but by a weighted average of the L and M cone response curves measured with color mixture matching.
These individual differences appear in all visual tasks, however they are measured. Peter Kaiser and Robert Boynton describe the situation this way: "Comparisons between different observers, whether in the same or a different experiment, present a discouraging picture. Although observers agree on certain major trends, individual differences are best described as enormous."
Standard psychometric methods are designed to disguise what is typically an "enormous" amount of individual variation in perceptual thresholds of detection and discrimination. These individual differences vary with the type of detection or discrimination required, on the specific way in which the visual stimulus is created and presented, and most importantly, on the perceptual and cognitive attributes of the observer, including past experience with the task it is possible to learn or practice your way to better visual acuity.
If you are curious about your visual performance as a visual astronomer, it will be very helpful to discuss testing for color normal vision, resolution and contrast sensitivity with your ophthalmologist. It is easy to measure pupil aperture yourself, when your eye is fully dark adapted. This information will provide an empirical basis for your assessment of telescopic optical performance.
Illuminance & Luminance
For the purposes of describing a light stimulus, the key distinctions are between illuminance and luminance.
Illuminance (E) is the quantity of light incident on a unit area of surface or passing through a unit area of aperture, defined as:
E = LI/d2
where LI is the luminous intensity of the light (the quantity of light radiated into a solid angle of 1 steradian, in lumens) and d is the distance in from the light source. When distance is in meters and the unit area is a 1 meter square surface or 113 cm diameter circular aperture, illuminance is measured in lux.
Luminance is the quantity of light emitted by a surface onto a surface, defined as:
L = E/s
where s is the solid angle of the light source (in steradians or square radians) as observed from the receiving surface or aperture. When both the light source surface and the receptor surface are standardized as 1 meter square, luminance is measured in candelas per square meter.
In optical systems, the objective focal length o and aperture area D2 and reciprocal relative aperture govern the relationship between light grasp and image luminance:
Lo = D2E/o2 = E/N2
As explained above, changes in relative aperture produce squared changes in image luminance.
The key distinctions between illuminance and luminance are:
Illuminance is completely invisible to the eye; all light perception is a form of imaged luminance, perceived either as an emitting light source or a reflecting material. All brightness and lightness perceptions are luminance perceptions.
Illuminance decreases as the light source is at a larger distance from the receiving surface. Luminance decreases as the same quantity of light is emitted by a physically larger source. Illuminance and luminance both decrease as the same amount of light is incident on a larger surface area.
Illuminance provides no information about the size, distance or intensity of the light source (the same quantity of illuminance can come from a nearby dim light or a distant bright light). Luminance is a specific measure of the visual geometry of the illuminance from a light source in relation to a light receptor; it depends both on the angular area (surface area and distance) of the light source and on the area of the light receptor (e.g., the diameter of the pupil), but is invariant with distance because a light source that decreases in illuminance with distance decreases in the same inverse square proportion in angular area.
Illuminance decreases with increasing distance between the light source and receiving surface. Luminance remains constant regardless of the distance between the emitting light and receptor surface the angular area of the source decreases with distance in the same inverse square proportion that the illuminance from the source decreases.
The power or light concentrating capability of an optical system is proportional to D2/2 = N2; a 2" aperture astrograph operating at /4 delivers more concentrated light to a photoreceptor than an /10 instrument of 20" aperture. Regardless of aperture, the luminance of an optical image is equal to the luminance of the source.
Illuminance and luminance are measurement units in photometry, which is limited to electromagnetic radiation that produces a response in the eye, usually considered to be wavelengths between 750 to 380 nm. Irradiance and radiance are the equivalent measurement units in radiometry, which includes all electromagnetic energy from radio waves through xrays. Thus, a star that radiates very powerfully but almost entirely in either the infrared or ultraviolet will appear visually faint, because little of the energy from the star can stimulate the eye.
Vision can be characterized by three adaptation states, each associated with a range of environmental incident light (illuminance) levels:
Scotopic. Only the rods respond to light, generally at illuminance levels below 0.1 lux. Several visual criteria define this adaptation, chief among them is the loss of color sensation, although memory color may intrude to tint familiar objects near the scotopic threshold.
Photopic. The cones predominate in the response to light, which nominally occurs at illuminance levels above 10 lux or more. In fact, the rods are still fully functioning at illuminance levels even 10 times higher, up to light levels typical of noon sunlight. Photopic adaptation characterizes the state in which color perception can be entirely described in terms of the cone sensitivities alone, and any increase in illuminance will not improve visual acuity (resolution of small angular intervals).
Mesopic. The transition adaptation between scotopic and photopic vision. Color perception is present, but color saturation and contrast are muted, and hues are somewhat biased by the rods.
Light adaptation is the response of the eye to increased luminance levels. It is generally prompt, occurring within a fraction of a second at the pupil and within less than a minute in the retina (via photopigment bleaching) and brain.
Dark adaptation is the response of the eye to decreased luminance levels. It occurs within 2 minutes or more, depending on the light exposure prior to darkness.
The astronomer is more interested in the dynamics of dark adaptation and the corresponding sensitivity of the eye to faint light in darkness. The common lore is that dark adaptation is best preserved by using a dim red light source in darkness, and this is borne out by dozens of research studies. There is no colored light source that is comparable to extreme red light, and most are equivalent to a white light. The main alternatives are red or white: white should be preferred only when it is necessary to perceive differences in color, for example in equipment electrical wiring or printed star charts.
Studies of the time required to recover from exposure to red or white light consistently show an advantage for red light, which becomes disproportionately greater if the light exposure is brighter than about 65 lumens (the intensity of a desk lamp). All light sources induce a greater recovery time (return to dark adaptation) as they become brighter, so regardless of lighting choice the first principle is to work in as dim a light environment as practicable.
It is sometimes claimed that full dark adaptation (recovery of the maximum sensitivity to low light levels) requires hours, days or even weeks to occur. This is entirely false. The photopigment rhodopsin is regenerated in a light shielded eye at the rate of 0.25% per second of the residual (unregenerated) quantity of bleached photopigment. This means, if 100% of the photopigment is bleached by exposure to an extremely bright light, that it would take 77 minutes for 99.999% of the bleached photopigment to be regenerated. This is effectively complete darkness adaptation. Performance measurements of visual sensitivity or acuity (diagrams, above, from Luria & Kobus, 1984) clearly indicate that dark adaptation after exposure to a relatively dim light source (not brighter than a desk lamp) occurs within about 10 minutes for white light and 4 minutes for red.
Again, a dim light level is as critical as a red light environment. Hecht & Hsia (1945) found that adaptation from a white light environment took 15 minutes (!) longer than adaptation from a red light environment if both lights illuminated at 3500 lux, but the white adaptation took only 2 minutes longer if both lights were at 30 lux. Hulbert (1951) measured the difference to adapt from 1070 lux at 14 minutes, and from 11 lux at 5 minutes. As most commercial incandescent "red" light bulbs actually emit a significant amount of "white" (broadband) light, the importance of the dimmest practicable light levels is obvious.
However, the astronomer does not reach the potential limit of dark adaptation, because the ambient "darkness" is typically much brighter. The physiological dark adapted threshold to light is around 0.000001 lux (reflected from a white surface), but the ambient illuminance from a "dark site" starry sky is about 0.001 lux and the light from a full moon is around 0.1 lux, five orders of magnitude greater than the absolute threshold. Suburban or periurban light pollution can also increase the sky brightness of a starry, moonless sky and raise the threshold that must be reached, reducing the recovery time needed to adapt to it.
Some research reports disparage as insignificant the difference in time required to recover from white versus red light exposure. The criterion used for this judgment is usually the time required to adapt only once from a subdued red or white light environment to complete darkness. The astronomer, alternately working at the eyepiece and with charts, guides, computer displays and notetaking or drawing materials, may make the transition from an illuminated to a dark visual environment several dozen times in a single evening. If the difference in adaptation between white or red light regimes amounts to only two minutes, this would result in a cumulative difference in adaptation times between red and white lighting of a half hour to an hour or more. This repetitive adaptation means that the most desirable lighting regime for the visual astronomer is the dimmest practicable red light, obtained solely from LED light sources, sufficient to perform the necessary visual tasks.
The light grasp of the eye as a function of aperture is quite modest. At a dark adapted pupil diameter of about 6 mm, the pupil area allows only 3.5% the light grasp of a 150 mm (6") telescope. However the eye is capable, under optimal conditions, of perceiving a light source emitting only a few photons per second. Thus, what it lacks in aperture it makes up in retinal sensitivity.
The minimum angular interval or linear dimension that can be resolved by a normal human eye under optimal viewing conditions varies with the type of resolution required.
The angular criterion is usually given as Rv = 60 arcseconds or 0.00029 radians for the resolution of two point light sources, and 120 arcseconds or 0.00058 radians for the resolution of parallel lines. A study by M.A. Danjon (1928) gave 74 arcseconds resolution for high (90%) contrast detail and 127 arcseconds for low (2.5%) contrast detail, which equals linear resolution criteria between 0.0897 to 0.1539 mm. The value 120 arcseconds seems to be commonly adopted, but resolution can clearly be better than that. My own resolution limit for the high contrast interference bands in a λ/D diffraction artifact is around 100 arcseconds.
The diagram (right) describes the eye's optical resolution in the abstract, as the spot diameter of a point light source on the retina as a function of pupil diameter, based on calculations with the Navarro ideal or mathematical "eye".
The Navarro eye allows the spot diameter to be separated into the contributions from the size of the diffraction artifact (the Airy disk, which gets smaller with larger pupil aperture) and from optical aberrations in the cornea and lens (geometric spreading, which increases with an eye pupil larger than ~3 mm). At the pupil sizes characteristic of most visual astronomy, the spot diameter in a normal eye is generally well above 25 μm (0.025 mm), which means the resolution of dim extended objects with an emmetropic, dark adapted eye (~6 mm pupil diameter) is roughly four times worse than the photopic resolution, in the range of 240 to 480 arcseconds (0.0012 to 0.0023 radians), and may be twice that when averted vision is used. (Note that a pupil diameter of around 3.5 mm would still apply to most lunar and some planetary observing in larger apertures, and that resolution values will generally be lower for older and visually impaired observers.)
The common method for estimating the eye's peak visual acuity is to calculate the average spacing between two cones in the fovea as the diameter of a single cone, and to assume that visual resolution is equal to this average spacing (the gap between two point stimuli is as wide as one cone). The average cone density is estimated to be around 190,000 per mm2 in the foveola (diameter on the retina of about 0.35 mm), and about 100,000 per mm2 in the fovea (diameter of about 1.85 mm). This works out to an average cone diameter of about 0.0025 mm in the foveola and 0.0034 mm in the fovea, given the characteristic hexagonal "tiling" of cone aperture.
With the eye's average internal focal length of 17 mm, these widths define an angular separation of between 30 to 41 arcseconds. However, the minimum spot diameter of the eye under optimal conditions (diagram, above) is about 7 μm or 0.007 mm, which is roughly the diameter of three cones. Thus the cone spacing method for predicting visual acuity overstates the eye's detection threshold under optimal conditions by at least two times and this is consistent with the measured performance acuity of the normal eye under bright light, which is between 1 and 2 arcminutes across the fovea. Thus, the limitation on the eye's visual acuity is primarily optical, not sensory, and optical performance strongly depends on the pupil aperture. Calculations based on retinal tiling are also imprecise because neural networks perform aliasing of the image that can actually exceed the theoretical resolution limit of the eye's optics.
Finally, the task dependence of visual acuity is reflected in the different discrimination thresholds that characterize different monocular visual tasks:
Shape recognition - Black on white letter shapes can be resolved (e.g., rotational orientations of the shape "E" can be identified) when elements of the letter are at least 5 arcminutes wide.
Grating resolution - Black on white line gratings (the lines and the spaces between them are of equal width) can be discriminated from a uniform gray background when the spacing between black lines is at least 2 arcminutes.
Point recognition - Two black points on a white background can be resolved when the space between them is at least 1 arcminute; white points on a black background can be discriminated at about 2 arcminutes (due to spreading of the image).
Vernier (Nonius) acuity - The misalignment between two black lines placed end to end or the vertical misalignment of two dots can be perceived when the lateral displacement is at least 10 arcseconds. This form of acuity is the principle behind the heliometer and the reason why it was the tool that first measured stellar parallax (Bessel and 61 Cygni).
Contrast or modulation range
The troland is a measure of retinal illuminance equal to the illumination from a surface emitting 1 lux per square meter into a pupil area of one square millimeter; it increases both with an increase in source brightness and an increase in pupil aperture. For comparison, 1000 trolands is approximately the brightness of a "white" paper viewed under bright indoor lighting, and 0.1 trolands is approximately the brightness of white paper viewed under moonlight.
This degradation of visual acuity with increasing pupil aperture is clearly indicated in the modulation transfer functions of normal eyes at various pupil apertures, measured by photographing the spot size actually imaged on the retina of subjects (diagram, right). For low contrast targets such as nebulae or faint planetary detail, the limit imposed by a large pupil
Field of View
Because the eye's normal point is the standard for visual magnification, calculation shows that the eye's true field of view is equal to its apparent field of view.
As a simple test of your true field of view: stand and look straight ahead with both eyes, then extend your arms on either side, fingers facing forward. Wiggle your fingers while moving both your arms forward and backward until you find the point where the movement of the fingers of both hands is just visible on both sides of your visual field. The angle between your arms will be about 120°.
However, the fact that your fingers are barely visible at 120° means that your useable field is much smaller. How small? At the minimum, the fovea is only able to resolve the central ~1° of your visual field. Perceptual research also shows that viewers will say a single object fills their visual field when it subtends about a 20° visual angle a generalization first made by Leonardo da Vinci. For example, the industry recommended wide screen TV viewing distances produce a visual angle to the wide screen diagonal of between 20° to 40°, with the average around 25° an image width that is said to produce "an immersive feel". By that standard, the 40° apparent field of traditional eyepieces would be fully immersive.
Currently available commercial eyepieces offer apparent fields of view between 40° to 100°. The choice of one end of that range over the other seems to tip at an apparent field of around 60°, and can depend on several considerations: esthetic preference (the "majesty factor" as opposed to "looking through a drinking straw"), visual task (sweeping for comets as opposed to resolving double stars), the telescope mounting and balance sensitivity (super wide field eyepieces can be very heavy, especially at long eyepiece focal lengths; super wide field eyepieces allow extended "drift" viewing of objects with hand guided Dobsonian altazimuthal mountings), objective relative aperture (off axis eyepiece aberrations are more noticeable at smaller focal ratios) and price (super wide field eyepieces are generally much more expensive than traditional designs).
Off axis image quality, in telescopes with focal ratios down to about /6, can be superior in some super wide angle eyepieces than in some traditional eyepieces, and image quality at the center of the field (on axis) is, for almost all oculars operating at all focal ratios, optically perfect. Only at objective focal ratios below /5 do optical considerations become important.
On the other hand, super wide field eyepieces require the observer to "look around" in the wide available field in order to see clearly, and this movement of the eye and head can cause vignetting or aberrations to appear. (Bias disclosure: I prefer eyepieces with apparent fields no greater than 70°, mainly to avoid these effects.)
In general, preference for wide or "narrow" apparent field eyepieces is not an issue of optical quality or the "immersive" visual area useful to the steady gaze of the human eye, but depends on the astronomer's personal preferences in visual esthetics, ergonomics and visual tasks, the convenience of use with a specific type of telescope ... and price.
Diopters are used to measure the distance that a lens must be moved forward or backward along the optical axis to produce a change in the image focus equal to one diopter of the eye. This focus diopter is found as:
Thus the movement required to adjust the focus of a 20 mm eyepiece by one diopter is (202)/1000, or ±0.4 mm.
Focus diopters may be positive or negative. At zero focus diopter, an eyepiece projects collimated light rays of light parallel to the optical axis, which can then focused by a relaxed normal eye as if the light came from a far distant source. Therefore the optimal focus diopter for a normal eye is zero or very slightly negative.
Astronomical Optics, Part 1: Basic Optics - an overview of basic optics.
Astronomical Optics, Part 2: Telescope & Eyepiece Combined - the design parameters of astronomical telescopes and eyepieces, separately and combined.
Astronomical Optics, Part 4: Optical Aberrations - an in depth review of optical aberrations in astronomical optics.
Astronomical Optics, Part 5: Eyepiece Designs - an illustrated overview of historically important eyepiece designs.
Astronomical Optics, Part 6: Evaluating Eyepieces - methods to test eyepieces, and results from my collection.
Amateur Astronomer's Handbook by J.B. Sidgwick - excellent basic chapters in optics, light gathering, resolution, magnification and more.
Telescopic Limiting Magnitudes by Bradley Schaefer - an attempt to predict telescopic limit magnitudes using visual data, star color, age of observer, sky brightness and other factors.
Human Eye from Handbook of Optical Systems, Vol. 4: Survey of Optical Instruments by Herbert Gross, Fritz Blechinger & Bertram Achtner (eds.). (Berlin, DR: Wiley-VCH, 2008).
Visual Acuity - Summary of the various methods used to test the human visus.
Average optical performance of the human eye as a function of age in a normal population by A Guirao, C González, M Redondo, E Geraghty, S Norrby and P Artal. Investigative Ophthalmology & Visual Science Jan. 1999, pp. 203-213.
The N.A.A. Telescope Calculator - handy and accurate calculator page for most optical parameters of a telescope/eyepiece combination.
Eyepiece Focal Length Measurement - Jim Easterbrook explains how to measure eyepiece focal lengths in grating projections.
The Collimation - an excellent optical discussion by Thierry Legault
Notes on LEDs - technical overview of the observatory light source most desirable for astronomers.
The Relative Effectiveness of Red and White Light for Subsequent Dark Adaptation (1984) - a useful literature review of the different effects of red and white lights on dark adaptation.
What Is a MTF Curve? - Basics of the MTF.
Last revised 11/26/13 ©2013 Bruce MacEvoy