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QUANTIFYING DESIGN CHOICES FOR THE COMPONENTS OF NEWTONIAN TELESCOPES by MAX E. KLINGER 3142 Folsom Street Boulder, CO 80304 Copyright 1992 by Max E. Klinger INTRODUCTION Traditionally, published advice on selecting the optical components of a Newtonian telescope has taken the form of a few rough guidelines for choosing among the welter of alternatives for focal length, diagonal size and the like. Often, these "guidelines" are little more than longstanding rules of thumb about particular aspects of the telescope. For example, there is the familiar suggestion to choose a fast focal ratio (F/5 or faster) for deep-sky observing and a slow focal ratio (F/8 or slower) for planetary work. These popular design guidelines tend to focus almost exclusively on selecting the F/ ratio, while virtually ignoring, in the design phase, consideration of other components, such as the eyepieces that will be used. Moreover, the size and placement of the diagonal are often treated more as discrete construction details to be worked out near the end of the process rather than as integral components of the overall design itself. The advent of modern eyepieces, with their wide fields and low powers, has highlighted the weaknesses in this approach. The variety of different eyepiece designs available today provides amateurs with a much wider choice and greater flexibility in the design of Newtonian systems, but exploiting this flexibility requires a more integrated or comprehensive approach than just selecting the F/ ratio of the primary mirror. Of course, such an approach also makes the task of choosing among the alternatives all the more difficult. In this context, however, one thing is clear. Approaching the design task by focusing almost exclusively on choosing the focal ratio for the objective mirror is rather like designing a vehicle, complete with its propulsion system, without determining what type of fuel will be used in it. A far better approach is to determine in advance how all the pertinent variables will interact. Doing this requires some amount of mathematical calculation, but with a computer and a suitable program, such as OPTIMAX, this can be done quite easily. Indeed, all of the calculations discussed in this booklet can be performed rather simply using the OPTIMAX program. The great advantage of such an approach is that one can rather effortlessly consider a large number of design alternatives and make well-informed choices among them. PERFORMANCE VARIABLES In terms of their physical components, four variables will generally determine the optical characteristics that a given Newtonian system will have: 1. The focal (or F/) ratio of the primary mirror; 2. The size of the diagonal mirror; 3. The placement of the diagonal, i.e., the distance between the center of the optical axis and the focal plane; and 4. The characteristics of the eyepieces that will be used. Independently, each one of these variables can have a profound affect on the optical characteristics of a Newtonian system, and the interplay among these variables will control not only the brightness of the visual images but also the relative size of the field of view. For most observers using a telescope for visual astronomy or astrophotography, these two features--the brightness of extended images and the field of view--probably more than any others, determine the suitability of a particular telescope for a given observer's interests. While there is some truth to the old guideline about choosing between relatively fast and slow focal ratios, this is only a small part of the story. With recent advances in eyepiece design and performance, the choice of focal ratio is somewhat less critical for amateur telescopes than previously. For example, today's wide field, low power eyepieces make it possible to achieve relatively wide fields and bright images (i.e., large exit pupils) with even moderate focal ratio (e.g., F/5 and F/6) mirrors. In fact, the eyepieces that will be used with the telescope play an extremely important part in determining both the relative brightness of deep sky objects as well as the telescope's field of view. In addition, the choice of eyepieces, coupled with the location of the diagonal, directly controls the size of the diagonal required. Since each one of these variables operates independently in determining the final optical characteristics of the system, it makes more sense to design a telescope by selecting the optimum configuration for all four of the variables in light of a specific observer's objectives. Especially with a small computer, this process can rather easily be significantly quantified. QUANTIFYING THE VARIABLES As Dennis DiCicco has noted in Sky & Telescope, April 1990, p. 358., the visual brightness of extended objects depends solely on a telescope's aperture and the magnification in use. For two telescopes of the same aperture, the relative visual brightness of the same extended image will be solely a function of the relative magnification. EXIT PUPIL Generally, the lower the magnification, the brighter the image. This can most easily be measured in terms of the exit pupil (the ratio of the size of the objective aperture in millimeters to the magnification). Exit pupil size = Size of the Objective / Magnification For example, in designing a visual telescope to produce wide field, bright images of deep sky objects, one should select a combination of objective focal ratio and eyepieces that produces the largest usable exit pupil, which is usually said to be about 6-7 mm. As Chart I shows, this combination is possible with a variety of combinations of different focal ratio mirrors and eyepieces. The exit pupil figures shown in Chart I below illustrate that the maximum useful brightness--exit pupils in the range of 6 to 7 mm--can be achieved with mirrors ranging in focal ratio from F/4 through F/8. In other words, comparably bright images can be produced with any mirror in this range, depending on the choice of eyepieces. If nothing else, this shows the limited value of the traditional approach for selecting telescope F/ ratios. ----------------------------------------------------------------- CHART I: Exit Pupil Sizes for Varying F/ ratio and eyepiece combinations with a 10" mirror Focal L. Eyepiece Magni- Exit F/ Objective F.L. fication Pupil Ratio (mm) (mm) (mm) 4 1016 18 56 4.5 4 1016 24 42 6.0 4 1016 32 32 8.0 5 1270 24 53 4.8 5 1270 32 40 6.4 5 1270 40 32 8.0 6 1524 24 64 4.0 6 1524 32 48 5.3 6 1524 40 38 6.7 7 1778 32 56 4.6 7 1778 40 44 5.7 7 1778 55 32 7.9 8 2032 32 64 4.0 8 2032 40 51 5.0 8 2032 55 37 6.9 Since it is possible to produce comparable exit pupils with a range of different focal ratios and eyepieces, one must then consider how these various choices affect the other components of the system. For example, the same exit pupil produced with different eyepieces and different focal ratios will affect two other components of the system, viz., the size and the location of the diagonal. DIAGONAL SIZE A given eyepiece will produce the desired effect only if it receives light from a diagonal large enough to illuminate adequately the field of the eyepiece. In turn, the required size of the diagonal is a function of both its location (i.e., its distance from the focal plane) and the size of the field that is to be illuminated. Generally, the larger the field to be illuminated, the larger the diagonal is required, while the greater the distance between the diagonal and the focal plane, the larger the diagonal is required. Size of the diagonal also affects two other characteristics of optical performance. The larger the diagonal, the more light it will prevent from reaching the objective mirror. In addition, the larger the diagonal, the more it will increase diffraction in the system and, hence, reduce contrast and resolution. Many years ago Allyn Thompson, Sky & Telescope, April 1945, p. 21. suggested that whenever the area of the obstruction caused by the diagonal exceeds five or six percent of the area of the objective mirror, the diffraction effects will become noticeable. The task then is compare how the various combinations of focal ratios and eyepieces will affect diagonal size and placement, as well as the extent of obstruction caused by the diagonal. To quantify the differences one must first determine the linear size of the image that is required for each eyepiece that is likely to be used. This can be approximated from the formula: i = L (AF / 59.29586°) where "i" is the linear image size in inches, "L" is the telescope focal length in inches, and "AF" is the size of the angular field of view in degrees. Since the angular size of the field is a function of the magnification and the apparent field of the eyepiece, one can calculate the linear image size that each eyepiece will produce. The size of the field is given by the formula: Real Field = Eyepiece Apparent Field / Magnification As is shown in Chart II below, the image size can be determined from the size of the field for various combinations of F/ ratios and eyepieces. The 50° apparent fields are typical of today's plössl-type eyepieces, while 65° fields are found in many of the better "wide field" eyepieces. Note that the image size for a given eyepiece does not vary with the focal ratio of the primary mirror. These sizes are given in English units, since, because diagonals themselves are usually sized in inches, it more convenient to calculate both the image size and diagonal size in inches. ---------------------------------------------------------------- CHART II: Image Sizes for Varying F/ ratio and Eyepiece combinations with a 10" objective mirror F/ Telescope Eyepiece Apparent Real Image Size ratio Fl (mm) Fl. Field Field (mm) (in.) 4 1016 18 50 0.9 15.7 0.6 4 1016 24 50 1.2 20.9 0.8 4 1016 32 65 2.0 36.3 1.4 5 1270 24 50 0.9 20.9 0.8 5 1270 32 65 1.6 36.3 1.4 5 1270 40 65 2.0 45.4 1.8 6 1524 24 50 0.8 20.9 0.8 6 1524 32 65 1.4 36.3 1.4 6 1524 40 65 1.7 45.4 1.8 7 1778 32 65 1.2 36.3 1.4 7 1778 40 65 1.5 45.4 1.8 7 1778 55 50 1.5 48.0 1.9 8 2032 32 65 1.0 36.3 1.4 8 2032 40 65 1.3 45.4 1.8 8 2032 55 50 1.4 48.0 1.9 ----------------------------------------------------------------- There are, in fact, a number of different ways to calculate the appropriate diagonal size. One traditional approach is to use the following formula: Diagonal size = l (D -b) / (F + b) where "l" is the distance from the center of the diagonal to the focal plane, "D" is the diameter of the primary mirror, "b" is the linear image size at the focal plane, and "F" is the focal length of the primary mirror. Many of the diagonal-size formulas traditionally recommended produce a diagonal that is often larger than necessary for telescopes designed for visual, photographic, or non-photometric astronomy. Some years ago, William Peters and Robert Pike, Sky & Telescope, March 1977, p. 220. developed a "minimal" diagonal approach based on the premise that complete illumination of a field to the very edge of the field is unnecessary for most purposes and that a certain amount of light loss, up to one-half magnitude, at the edge of the field is unnoticeable. According to Peters and Pike, most photographic lenses, for example, display at least this much light loss at the edge of the field. Although this approach is computationally complex, with a small computer the task can be made manageable. The following analysis relies upon the Peters-Pike method. Mathematically, the smallest possible size for a diagonal mirror is given by the formula: Diagonal size = D (l / F) where "D" is the diameter of the primary mirror, "l" is the distance from the center of the diagonal (the optical axis) to the focal plane, and "F" is the focal length of the primary mirror. Although most texts usually recommend a diagonal size significantly larger than that given by this formula, Peter and Pike suggest using this smaller formula unless such a diagonal produces a light loss at the edge of the field of more than .5 magnitudes. In those cases, a larger diagonal is used to make sure that the light loss at the edge of the field is no greater than .5 magnitudes, which for most non-photometric purposes would not be noticeable. Peters and Pike determine the degree of illumination at the edge of the field (I) from the following equation: I = (arccos A - x√(1- A) + r² arccos B) / π where the subsidiary quantities are given by the following equations: r = aF / lD x = 2b(F - 1)/ lD A = (x² + 1 - r²) / 2x B = (x²+ r² - 1) / 2xr in which "a" is the size of the diagonal's minor axis, "F" is the telescope's focal length, "l" is the distance from the optical axis to the focal plane, "D" is the diameter of the primary mirror, and "b" is the linear size of the image at the focal plane. To express the loss in illumination in magnitudes the formula 2.5 log 1/I is used. Since diagonal size varies with the distance from the diagonal to the focal plane, changing this distance, for example, by changing the height of the focuser, provides a range of design options. Chart III shows that for a 10 inch F/5 system, varying the diagonal-to-focal plane distance from 8 inches to 13 inches reduces the size of the diagonal by about .4 inch, which represents about a 20% reduction in diagonal size. ----------------------------------------------------------------- Chart III: Diagonal Size as a function of distance to focal plane (for 10" F/5 system (1.75" image) Distance to Diagonal Focal Pl. Size 8 2.04 9 2.10 10 2.18 11 2.25 12 2.34 13 2.43 ----------------------------------------------------------------- Diagonal size is also affected by the linear image size. This variation can be seen in Chart IV below, which shows that changing the image size from .5 inch (such as for a 12-18 mm plössl eyepiece) to 1.75 inches (suitable for 35 mm photography) increases the size of the required diagonal by 66% in the case of an F/5 system and 34% in an F/6 system. ----------------------------------------------------------------- Chart IV: Diagonal Size as Function of Linear Image Size (for 10" Mirror) F/5 System F/6 System Image Size Diagonal Diagonal 0.50 1.21 1.50 0.75 1.80 1.50 1.00 1.80 1.50 1.25 1.80 1.65 1.50 1.94 1.82 1.75 2.11 2.01 ----------------------------------------------------------------- If commercially available diagonals are to be used, the range of options is somewhat constrained, since diagonals are usually available in only about 8 or 10 discrete sizes within the range of .5 inch to 4.25 inches. In practice, using commercial diagonals may mean either making a compromise to accept a larger or smaller diagonal than is the optimum or adjusting other components in the system to bring the optimum diagonal size closer to the available size. For example, the optimum diagonal for a 10 inch F/5 system with a 1.75 inches image size and an 11 inches distance from the diagonal to the focal plane is 2.26 inches. The closest standard diagonal size is usually about 2.14 inches, whereas the next larger size is usually 2.6 inches. Using the 2.14 inch diagonal would, under the Peters-Pike method, result in only 59% illumination or a .57 magnitude loss of light at the edge of the field, whereas the 2.6 inch diagonal would produce 75% illumination and .32 magnitude loss at the edge of the field. The 2.6 inch diagonal, however, would also represent an area of about 6.7% of the primary mirror, as opposed to 4.6% for the smaller diagonal. A diagonal of 2.6 inches might well produce noticeable diffraction effects and a loss of contrast. The traditional wisdom was that, when faced with such a choice, one should select the smaller diagonal if the telescope will be used primarily for visual work. See Allyn Thompson, Sky & Telescope, April 1945, p. 21. A less drastic alternative is reducing the distance from the diagonal to the focal plane. In this instance, reducing the distance from 11 inches to 10 inches reduces the diagonal to 2.21 inches, while a further reduction to 9 inches results in a diagonal size of 2.11 inches, which closely corresponds to the 2.14-inch size commonly available. Another option is to accept a smaller image size, perhaps by changing the lowest power eyepiece, which would also reduce the size of the diagonal needed. A QUANTIFIED DESIGN APPROACH The foregoing examples illustrate that, in many instances, the task of selecting the components for a Newtonian system can best be accomplished by taking an approach exactly backwards from that traditionally recommended. Instead of starting with the primary mirror, start at the opposite end of the system -- with the eyepiece! By selecting first the size of the real field and the brightness (or exit pupil) one desires the telescope to produce, it is possible then to determine what combinations of eyepieces and focal ratios will produce the desired results. This in turn indicates the linear size of the image that will be needed, which in turn determines the range of diagonal sizes and placements required to illuminate properly the field of the desired size. Performing the calculations outlined here would ultimately provide a set of different combinations of F/ ratios, eyepieces, diagonal sizes, and placements that will approximate the desired results. For example, starting with the goal of building a wide field, low power 10 inch Newtonian for visual, deep-sky work, one might arrive at the sets of combinations shown in Chart V, below. ----------------------------------------------------------------- Chart V: Image Sizes for 10" Mirror F/ Eyepiece Apparent Mag. Real Exit Image Focal L. Field Field Pupil Size 4 24 50 42 1.2 6.0 0.82 5 32 50 40 1.3 6.4 1.10 6 32 65 48 1.4 5.3 1.43 6 40 65 38 1.7 6.7 1.79 8 40 65 51 1.3 5.0 1.79 8 55 50 37 1.4 6.9 1.89 ----------------------------------------------------------------- Each of the combinations shown in Chart V would, in turn, require the diagonals shown in Chart VI (assuming a diagonal to focal plane distance of 10 inches, which would put the focus about 4 inches outside a 12 inch tube, which would be about right for a tall 2 inch diameter focuser). ----------------------------------------------------------------- Chart VI: Diagonal Sizes 10" Mirror; Focal Plane Distance = 10" F/ Exit Image Diagonal % Area Standard % Area Pupil Size Size Diagonal 4 6.0 0.82 2.50 6.25 2.60 6.76 5 6.4 1.10 2.00 4.00 2.14 4.58 6 5.3 1.43 1.67 2.78 1.52 2.31 1.83 3.35 6 6.7 1.79 2.09 4.45 2.14 4.58 8 5.0 1.79 1.97 3.88 2.14 4.58 8 6.9 1.89 2.05 4.20 2.14 4.58 ----------------------------------------------------------------- The required diagonals for these systems would, using commercial diagonal sizes, range from 1.52 inches to 2.6 inches, which represents a range of 2.31 % to 6.76 % of the area of the primary mirror. Of course, one might also try reducing the diagonal to focal plane distance, which will affect the size of the required diagonal. This range of alternatives allows one to make a variety of choices. Since all of these systems will produce exit pupils in the range of 5.0 mm to 7.0 mm, they should produce comparably bright images. Similarly, the real fields for these systems, with one exception, vary by only a few tenths of a degree, which is not likely to be noticeable for most purposes. The choice among these systems, however, has a number of other consequences. For example, producing the 6.9 mm exit pupil and 1.5° field in the F/8 system would require using a comparatively expensive 40 mm wide field eyepiece that would need a 2 inch diameter focuser. Similarly, the physical size of the system, with its longer focal length and tube, would be bulkier and heavier. On the other hand, the F/8 system would probably also produce a larger coma-free field. By comparison, the F/6 system with a 32 mm 65° eyepiece would produce a slightly smaller exit pupil (5.3 mm) and approximately the same size field (1.4°), but it would also require a significantly smaller diagonal. A 1.83 inch diagonal would be only 3.35% of the area of the primary, which could translate into reduced diffraction effects and better contrast. While no amount of calculation can substitute for high quality optical and mechanical components, quantifying the relationships among the various components can provide a good foundation for choosing wisely among the myriad design choices available today. PUTTING IT ALL TOGETHER While the approach suggested here may seem rather complex, in fact with a suitable computer program like OPTIMAX the whole process can be reduced to a fairly straightforward process of experimenting with different options until the desired goal is achieved. The process described above, which would be essentially the same whether designing a telescope from scratch or upgrading an existing telescope, can be outlined as follows: 1. Select a set of eyepieces, either those that you already own or are would consider purchasing for your telescope. For this hypothetical set of eyepieces, determine what results (i.e., real field, magnification, and exit pupil) they would produce with a given mirror and F/ ratio. Since many manufacturers produce both regular eyepieces and wide-field eyepieces in about the same range of focal lengths, it is possible to vary the magnification, the size of the real field, and the exit pupil independently by trying different combinations of focal lengths and eyepiece apparent fields. Those trying to select among several possible mirrors or F/ ratios for their system will want to consider several sets of results, one for each different mirror or F/ ratio. Many manufacturers will supply information about the apparent fields of their eyepieces. If such information is not available, a rough estimate can be made based upon the following listing of the typical fields for various eyepiece designs (adapted from R. Berry, Build Your Own Telescope p. 180 (Charles Scribners Sons 1985): 1. So-called "Ultra Wide-field" eyepieces, including Nagler types -- 82° 2. So-called "Wide-field" eyepieces, including Erfle-types -- 55°-65° 3. Plössl-type eyepieces -- 45°-50° 4. Orthoscopic eyepieces -- 45° 5. Kellner-type eyepieces --typically about 45° 6. Ramsden eyepieces -- 35°-40° 7. Huygens eyepieces -- 25°-35.° Of course, there are fairly precise observational techniques for determining field sizes that are described in many books on observational astronomy. 2. Determine the Diagonal Size. Based upon the image size for largest field-lowest power eyepiece that you plan on using, determine the optimum diagonal size. In doing this, you should experiment with different diagonal placements. Changing the location of the diagonal (i.e., locating it at different points along the line from the mirror to the focal plane) will affect the size of the diagonal. This is particularly important when using commercial diagonals, since the number of different diagonal sizes is somewhat limited. The OPTIMAX program automatically determines the results with the most commonly available diagonal sizes. In any case, if the optimum diagonal size differs significantly from the standard sizes available, one option is to experiment with changing the location of the diagonal to bring the optimum diagonal size closer to one of the available diagonals. In changing the distance between the diagonal and the focal plane, keep in mind the size of tube that you will be using as well as the height of the focuser. Obviously, having the image focus outside the focuser tube is to be avoided! Commercially available focuser heights can range from 1 or 2 inches to 4 or 5 inches. This range provides considerable flexibility in adjusting the diagonal size and placement. After determining the appropriate diagonal size, one must then consider the size of the diagonal, or more precisely the area of the diagonal, in relation to that of the primary mirror. Generally, the smaller the diagonal as a percentage of the area of the primary, the better. Certainly, diagonals whose area is more than 5% or 6% of the area of the primary should, if possible, be avoided, unless one is prepared to accept the likely resulting loss of contrast and increase in coma. If the optimum diagonal size for one configuration is "larger" than desired, try changing one or more of the other variables, such as the location or the image size, until a more suitable diagonal size will work. USING OPTIMAX As supplied, OPTIMAX comes with data file called OPTIMAX.WKB which contains sample entries and calculations for 10" F/5 Newtonian. The eyepiece chart shows the data for a cross-section of eyepieces with a 10" F/5 telescope, while the Automatic Diagonal Chart and the Manual Diagonal Chart both show calculations of the optimal diagonal size for a 10" F/5, with an image size of 1.1" (which is the image size required for a 32 mm., 50° plossl-type eyepice, and for a diagonal-to-focal plane distance of 10". To preserve this set of data for future reference, be sure to save any new data entered under a new *.WKB file. Using the "/fg" command ("file-get") you can load different *.WKB files into OPTIMAX. The basic operation of the program consists of four different components. 1. Eyepiece and Image Size Chart 2. Diagonal Size Chart (Automatic Mode) 3. Chart Showing Relative Area of Diagonals 4. Diagonal Size Chart (Manual Mode) (Component # 4 is an alternative to ## 2 & 3). 1. Eyepiece and Image Size Chart This chart will determine the four characteristics of eyepiece performance shown in the four columns on the right side of the screen for up to 21 different eyepieces. To use, insert the size of the objective mirror (in inches) and the focal ratio at the top of the chart and the focal length and apparent field size for each eyepiece in the two left columns. When this data has been entered, entering the information called for in the first two columns on any line will cause the characteristics in the last four columns to be calculated automatically. To determine the effects of using a barlow lens, instead of entering the telescope's normal F/ ratio at the top of the chart, enter the F/ ratio multiplied by the barlow's magnifying power. For example, in the case of an F/5 with a 2.5x barlow, enter "5*2.5" or 12.5 as the focal ratio. The data shown for the eyepieces will be correct for the 2.5x barlow. 2. Diagonal Size Chart (Automatic Mode) In this mode, the program will make a series of determinations about the optimum diagonal size as well as about the performance of various readily available commercial diagonals. To use this feature, enter the objective mirror size in inches, focal ratio, desired linear image size in inches, and the desired distance from the center of the diagonal (optical axis) to the focal plane (in inches), and then have the program calculate the diagonal size. The method of calculating the diagonal size is based upon the method described by William Peters and Robert Pike in Sky & Telescope, March 1977 p. 220. Under this method, the "% Illumination" represents the amount of vignetting at the edge of the field as a percentage of full illumination. When this percentage is 63%, it represents a .5 magnitude light loss at the edge of the field, which is usually unnoticeable for most purposes under the Peters-Pike method. An asterisk in the column marked "Unsuitable Diag." means that the diagonal size shown is smaller than the theoretical minimum size diagonal. For diagonal sizes or F/ ratios that are at the extremes for a given configuration, the formulas used for these calculations become indeterminate. This has several consequences. For extreme F/ ratios (e.g., smaller than F/3 or larger than F/12) with very small mirrors, the formulas may produce odd results. Similarly, for diagonal sizes that are much larger or smaller than the optimum size, the results may occasionally be spurious. None of these problems arises with reasonable and usual F/ ratios and diagonal sizes. When the results for a given diagonal size appear as "Indeter. Large" this indicates a diagonal size that is smaller than the optimum size, and the formulas indicate that the % of Illumination of the field is negligible while the magnitude loss becomes extreme (i.e., indeterminately large). Similary, when the results for a given diagonal size appear as "Indeter. Small," this indicates a diagonal size that is larger than the optimum size, and the formulas indicate that the % of Illumination of the field is essentailly 100% while the magnitude loss is indeterminately small. 3. Chart Showing Relative Area of Diagonals After using part 2, go to the Chart in Part 3. This will show, for the diagonals determined in Part 2, the relative area of each diagonal in relation to the size of the objective mirror Optimally, a diagonal larger than about 5-6% of the area of the objective mirror may increase diffraction and cause a loss of contrast. 4. Diagonal Size Chart (Manual Mode) This part allows the manual determination of the performance of a given diagonal size or a manual search for the optimum diagonal size. To use this feature, enter the objective mirror size in inches, focal ratio, desired linear image size in inches and the desired distance from the center of the diagonal (optical axis) to the focal plane (in inches). Then enter a diagonal size in the space shown and recalculate. The percentage illumination and magnitude loss (as explained above) as well as the relative area of the selected diagonal are shown. To determine manually the optimum diagonal size under the Peter-Pike method mentioned above, keep substituting different diagonal sizes until the magnitude loss closely approximates .5. Of course, using the manual mode one can also determine the smallest diagonal size that will fully illuminate the eyepiece field. To do this, try different diagonal sizes until 100% illumination is achieved. The warning that a "diagonal is too small" means that the diagonal size entered is smaller than the theoretical minimum, without regard to considerations of vignetting. Under no circumstances would such a diagonal be suitable, no matter what magnitude loss that is shown. In using this chart, it is important to remember the limitations imposed by the mathematics used for these calculations. These are discussed above in Point 2.