Introduction: The Question and My Approach...  Exit Pupil as A Criterion for the Selection of Focal Lengths for Eyepieces  My Own "Recommendations"  References  Appendix: Exit Pupil  Appendix: Magnification  Appendix: Field of View
In progress
On this and further pages, I will deal with the selection of suitable eyepiece focal lengths for your own (or planned) telescopes as well as with the assessment of whether existing (or planned) eyepieces are suitable for these telescopes and fit together. Starting point are recommendations on the basis of the size of the exit pupil, which I found in the literature or on the Internet and which I have "consolidated" in my own recommendations.
Once again, I would like to emphasize that I am still an astronomy beginner who has found these criteria in the literature or on the Internet, and now tries to apply them to his own equipment, hoping that this information is useful for other beginners in astronomy, too. I am far from giving recommendations for specific eyepieces, because I lack the respective experience. But be assured that the Internet is full of such recommendations ...
Note: For definitions in a small glossary, see page Quick & Dirty Astronomy Glossary.
When you as a beginner purchase an entrylevel telescope, this is often supplied with one or two eyepieces. Their quality is, however, often enough insufficient, so that one would like to acquire a set of new eyepieces, either quickly, or in the course of time. One is then faced with the problem of finding eyepieces with focal lengths that make sense and fit together. And one wonders whether there are any general criteria that allow statements about how a meaningful set of three or more eyepieces should look like. In books and on the Internet, I came, more or less accidentally, across the approach of using the size of the exit pupil as a criterion for choosing focal lengths of eyepieces. I found three recommendations and would like to introduce these in the following.
However, because these proposals partly "overlap" and partly complement each other, I found it difficult to deal with this diversity in practice. Therefore, I derived my own "consolidated" recommendations from the three recommendations and present them further down on this page.
Please note that on this page I present my "knowledge acquisition process," which serves as a "justification" for my recommendations and is therefore important, at least or perhaps, only for me. Many readers may, however, only be interested in the final result and its practical application. These readers should skip this page and go directly to the next page, where I briefly repeat my recommendations and apply them exemplarily to my telescopes and  "afterthe fact"  to my existing eyepieces. Readers should have no trouble applying the recommendations to their own telescopes (and retrospectively to their own eyepieces). All they need is the focal ratio (for calculating the focal lengths of the eyepieces) and the aperture (for calculating the magnifications) of their own telescopes!
In the discussion that follows, I will use a number of terms without going into them further. Definitions and further information about these terms can be found in appendices. If you like, you can read more there about:
I found three recommendations that suggest, which exit pupil diameter is suitable for which (deep sky) objects / uses. From this data you can calculate the focal length of the eyepiece and the magnification (see the appendix for the formulae). Please note that in the following formulae using the exit pupil, the focal length of the eyepiece just depends on the focal ratio of the telescope, whereas the magnification depends on the aperture of the telescope.
Note: If you do not want to read the recommendations in detail, you can jump directly to the summary or even to my recommendation that I derived from these three recommendations. Even faster you go with jumping to the next page, where I repeat and apply my recommendation!
In his book Deep Sky Reiseführer, Ronald Stoyan provides criteria for the selection and calculation of the focal lengths of eyepieces; these are presented in the table below. Actually, he provides formulae to calculate the magnification, but he does not give a justification for his recommendations and why he uses the exit pupil as a criterion (he even does not mention explicitly that the numbers in the denominators correspond to the exit pupil). I added the column for the calculation of the focal length of the eyepiece, because that is the value that is needed.
Deep Sky Application Area  Exit Pupil (mm)  Calculate Focal Length of EyePice from* 
Calculate Magnification from 
Search of objects, largearea nebulae  7 
Focal Ratio * 7 
Aperture / 7 
Nebulae, star clusters  4 
Focal Ratio * 4 
Aperture / 4 
Galaxies, globular clusters  1.5 
Focal Ratio * 1.5 
Aperture / 1.5 
Planetary nebulae, small galaxies  0.7 
Focal Ratio * 0.7 
Aperture / 0.7 
Double stars, small planetary nebulae  0.4 
Focal Ratio * 0.4 
Aperture / 0.4 
*) Added by me
Stoyan emphasizes the importance of extending the range of magnifications to the minimum and maximum magnifications. However, according to Stoyan, with small focal ratios (f15  f8), the minimum magnifications cannot be realized because there are hardly any eyepieces with focal lengths of more than 40 mm available.
In a discussion thread, I found similar, but slightly simpler suggestions for the eyepiece selection from poster penumbra (again, without a justification). He specifies only three applications (corresponding to 3 eyepieces), but he provides values ranges for the exit pupil. I added the columns for the calculation of the focal length of the eyepiece and the magnification.
Deep Sky Application Area  Exit Pupil (mm)  Calculate Focal Length of Eyepiece from* 
Calculate Magnification from* 
Maximum overview for largearea nebulae  4.5  6 
Focal Ratio * 4.5...6 
Aperture / 4.5...6 
Galaxies and midsize deep sky objects  2  3 
Focal Ratio * 2...3 
Aperture / 2...3 
Maximum magnification for moon and planets  0.6  1 
Focal Ratio * 0.6...1 
Aperture / 0.6...1 
*) Added by me
On the page on eyepiece consultation offered by Intercon Spacetec of the televue.de Website and on a newer page with the same content directly from Intercon Spacetec, I found information about the size of the exit pupil, which can also be transormed into a recommendation on the basis of exit pupil sizes. The pages also contain many useful hints, so that I strongly recommend to visit one of the two original pages (new, old).
I put the for me relevant content of the pages it into the following table format, so that it roughly corresponds to the above tables:
Deep Sky Application Area  Comment  Exit Pupil (mm)  Calculate Focal Length of Eyepiece from*  Calculate Magnification from*  
Maximum Field of View  Finder Function/Search  Especially with large telescopes an opening loss can be accepted, as long as one uses this consciously for searching and finding, and not for observing the objects.  6  10 
Focal Ratio * 6...10  Aperture / 6...10 
Minimum Magnification / Large Field of View  Overview  Given by the size of the pupil of the human eye  4  6 (5**; max. 8) 
Focal Ratio * 4.5...6 
Aperture / 4.5...6 
Normal Magnification 

2  4 mm exit pupil: According to the author, these eyepieces are the most frequently used ones.  3.5  4 
Focal Ratio * 3.5...4 
Aperture / 3.5...4 
2 
Focal Ratio * 2 
Aperture / 2 

Maximum Magnification / Maximum Resolution 

1 mm down to min. 0.8 ... 0.5 mm exit pupil

1 
Focal Ratio * 1 
Aperture / 1 
0.8 
Focal Ratio * 0.8 
Aperture / 0.8 

0.5 
Focal Ratio * 0.5 
Aperture / 0.5 
*) Added by me; **) the author writes (translated): "Unless the largest possible field of view is to be achieved, I consider an exit pupil of approx. 5 mm to be perfectly sufficient. An exit pupil of more than 5 mm does produce a brighter image. However, this only makes sense when the night sky is absolutely dark, and for the perception of some lightweak objects it only brings a slight increase to the absolute maximum.
Note: The names for the magnifications were taken from the original page. On page Magnification, I try to "sort" the terms and assign them to values of the exit pupil and the resolution.
When looking at the three recommendations, one finds a number of similarities, but also differences. I tried to arrange the three recommendations in a common table so that it is easier to compare them:
Intercon Spacetec

Stoyan

penumbra


Deep Sky Application Area  Exit Pupil (mm)  Deep Sky Application Area  Exit Pupil (mm)  Deep Sky Application Area  Exit Pupil (mm)  
Minimum Magnification / Maximum/Large Field of View  Search  6  10 
Search of objects, largearea nebulae  7 

Overview  4  6 (5; max. 8) 
Maximum overview for largearea nebulae  4.5  6 

Normal Magnification  Optimal for largearea, faint nebulae  3.5  4 
Nebulae, star clusters  4 

Perceptibility optimal for many objects, e.g. for most galaxies  2 
Galaxies, globular clusters  1.5 
Galaxies and midsize deep sky objects  2  3 

Maximum Magnification / Maximum Resolution  Actually the "normal" upper magnification limit...  1 
Maximum magnification for moon and planets  0.6  1 

With perfect seeing, achieves maximum perceptibility of small, lowcontrast details; maximum magnification for planets that makes sense  0.8 
Planetary nebulae,
small galaxies 
0.7 

Only usable for separating narrow double stars, and, at the extreme limit of the telescope, to perceive the faintest details.  0.5 
Double stars, small
planetary nebulae 
0.4 
Further below, I try to create a table with "personal recommendations" from this table. See my comments there, why I do so...
Originally, I had calculated eyepiece focal lengths for each of the recommendations that I had found and also looked at how my existing eyepieces fit. Then, there were just two recommendations, but in the meantime, there are three (see above). Therefore, another approach seems to me to make more sense to me, namely to derive my own "recommendations" from them and to calculate eyepiece focal lengths only for these or to check the existing eyepieces against these. This will happen on the following page! Here is, first of all, my "consolidation" of the three recommendations into a version of my own, in which I also included some thoughts from the preceding section Additions:
Category  Deep Sky Application Area  Exit Pupil (mm) 
Minimum Magnification / Maximum/Large Field of View  Search  7...10 
Overview, largearea nebulae  4.5...5...6 (7) 

Normal Magnification  Optimal for largearea, faint nebulae; nebulae, star clusters  3.5...4 
Perceptibility optimal for many objects, e.g. for most galaxies, globular star clusters, and midsize deep sky objects  (1.5) 2...3 

Maximum Magnification / Maximum Resolution  Actually, the "normal" upper magnification limit...  1 
With perfect seeing, achieves maximum perceptibility of small, lowcontrast details; planetary nebulae, small galaxies; maximum magnification for planets that makes sense  0.6...0.7...0.8 

Separation of narrow double stars, small planetary nebulae; at the extreme limit of the telescope, to perceive the faintest details  0.4...0.5 
Maybe, I should have divided the first item "Minimum Magnification / Maximum/Large Field of View" into the two items "Maximum Field of View" and "Minimum Magnification / Large Field of View," but that would only be "cosmetics." Note that the first item can lead to long focal lengths, which can only be reasonably used with 2" eyepieces (if such eyepieces exist at all...).
The third item "Maximum Magnification / Maximum Resolution" takes into account that the focal lengths of the eyepieces should be finer graded at high magnifications in order to adapt the focal length of the eyepieces to the effects of Seeing.
This ends the "theoretical" part of this topic, and I will turn to applying this knowledge on the following page...
Now, all you have to do is multiply the exit pupil values of the recommendation with the focal ratio(s) of your telescope(s) to arrive at the focal lengths of the eyepieces that are suitable for your telescope(s). I will demonstrate this step exemplarily for my own telescopes on the next page!
The exit pupil of an eyepiece, the definition is given below, is calculated from:
This leads to a formula for the focal length of the eyepiece, which is what we actually need when selecting eyepieces:
In practice, this means that the corresponding focal lengths for eypieces can be calculated from values for the exit pupil, as they are typically given in the recommendations for the focal lengths of eyepieces AND from the focal ratio of the telescope at hand. And not just for one telescope, but for all telescopes having the same focal ratio, no matter how large or small they are.
Alternatively, the exit pupil diameter is given by aperture of a telescope divided by its magnification (see below for details):
Now to the definition of the exit pupil!
In optical devices for direct visual observation  e.g. telescopes and binoculars  the exit pupil is the diameter of the ray bundle that leaves the eyepiece (according to Wikipedia).
In the end, only the size of the exit pupil determines how bright the image of a certain object, for example the moon, appears in the eyepiece. If the exit pupil is the same, the object always appears equally bright, regardless of the telescope, aperture, and magnification.
The exit pupil of a telescope should be matched to the pupil, better to the entrance pupil, of the human eye. After dark adaptation, the pupil is about 7 mm wide for young people and about 6.4 mm wide for adults; it gets smaller with growing age. An exit pupil of 6 mm is, according to the German Intercon Spacetec Website, however, still valid for observers that are 70 years old.
General rules for the exit pupil are:
The exit pupil determines (in the direction of the eye) the minimum as well as the maximum usable magnification of an optical instrument, that is, of a telescope. (From Wikipedia)
Stoyan therefore emphasizes that it is important to extend the range of magnifications to the minimum and maximum magnifications when choosing focal lengths of eyepieces. However, according to Stoyan, with small focal ratios (f15  f8), the minimum magnifications cannot be realized because there are hardly any eyepieces with focal lengths of more than 40 mm available.
After I had created an Excel table with magnifications for my telescopes and, using these magnifications, in a second table the focal lengths of the eyepieces for my telescopes, I realized that this was a somewhat cumbersome approach. After some transformations of the formula for the exit pupil, I found that the focal ratio is actually the "critical" telescope parameter for determining the focal lengths of eyepieces (if it is unknown).
I start from the formula for the exit pupil::
If the formula is solved according to the focal length of the eyepiece, you get:
You get the magnification by inserting the focal length of the eyepiece into to following formula:
Stoyan and other author use a different approach. For illustration purpose, I start from the same formula as above and insert the formula for the magnification:
Solving the formula for the magnification, results in Stoyan's formula:
For determining the focal length of the eyepiece, you can insert the formula for the magnification and solve the formula for the focal length of the eyepiece:
Ultimately, both paths lead to the same end, because, in addition to the focal length of the eyepiece, one needs to know the magnification to be able to determine whether an eyepiece's focal length makes sense (i.e., whether the magnification lies within the range of minimum and maximum usable magnification).
*) Or simply "aperture" as Stoyan puts it
The magnification of a telescope can be calculated as follows:
For a given telescope with a specific focal length, magnification depends only on the focal length of the eyepiece in use and can be easily calculated in your head.
The formula above suggests that any magnification can be achieved. In fact, however, there are limits at the top and bottom of the range, which I discuss in more detail on page Magnification. If one disregards certain influencing variables, such as the quality of the sky or the practical availability of certain eyepiece focal lengths, and focuses on the exit pupil criterion, one can say (already mentioned above...) that the exit pupil to the eye limits both the minimum and maximum usable magnification of a telescope. Stoyan therefore explicitly points out in his recommendations that the eyepiece set should fully exploit this range. Below the minimum magnification, the socalled "search function" can be useful.
In the following some remarks about the viewfinder function, minimum magnification, maximum magnification, eyepiece gradation at high magnifications, and about the use of magnifications in practice!
The viewfinder function uses a magnification smaller than the minimum magnification and refers to eyepieces with an exit pupil of 6 (or 7) to 10 mm, that is, eyepieces with an exit pupil in which light is wasted because the light beam leaving the eyepiece is larger than the eye pupil. Correspondingly, the field of view is large, and the magnification is low. In this function, the eyepiece is consciously used to locate objects and not to observe them, and light losses are no or only a minor problem. (According to Intercon Spacetec Website)
Examples
Stoyan's note on minimum magnification applies also to the viewfinder function (see next section), namely that there are hardly any eyepieces with a focal length of more than 40 mm. This is especially true for 1.25" eyepieces, but there are not many 2" eyepieces with focal lengths greater than 40 mm as well.
At the minimum magnification, Stoyan's above cited requirement is automatically met because it is calculated directly from the aperture ratio of the telescope and the maximum possible exit pupil. It should be noted, however, that the pupil value of 7 mm assumed by Stoyan is only valid for young people; for adults I found a value of 6.4 mm, and it should decrease further with age. However, according to the Intercon SpacetecWebsite, active observers, even at the age of 70, should not consider themselves below 6 mm (according to the Gahberg study). On the other hand, the author considers a pupil value of 5 mm to be quite sufficient if the largest possible field of view is not to be achieved. In any case, the range of 4.5 to 6 mm that penumbra indicates appears to be more appropriate in reality than a value of 7 mm.
Stoyan points out that for telescopes with a large aperture ratio (F/8 ... F/15), the calculated minimum magnification can no longer be achieved, because there are hardly any eyepieces with a focal length of more than 40 mm. This is particularly true for 1.25" eyepieces, but there are even few 2" eypieces with a focal length greater than 40 mm. This can perhaps be "adapted" a little by assuming a smaller exit pupil (6.4...4), but only marginally...
Another problem is that with 1.25" eyepieces having a focal length of 32 mm or more, viewing is always difficult. I made this experience with a 40 mm Plössl eyepiece, a starfriend also reported this to me, and a dealer confirmed this as well. In this case, it is good if you have a telescope for 2" eyepieces at your disposal, and to use it with 2" wide field eyepieces (for example, I own a 2" eyepiece with a focal length of 35 mm).
Examples
On the page Magnification, I discuss the various definitions of magnification, including terms such as normal magnification, beneficial magnification, maximum useful magnification, and maximum magnification. It is difficult for me to understand all the different definitions by different sources, but all of them can be expressed as multiples of the telescope aperture in mm. The maximum magnification defined by Stoyan is 3 x the aperture in mm, the maximum usable magnification, which also goes under different names, is 2 x the aperture in mm, and the beneficial magnification amounts to 1.5 x the aperture in mm. The Stoyan's maximum magnification is therefore twice the beneficial magnification!
Since it is easy to determine the values for these magnification types for a given telescope, you can do this quickly and determine the focal lengths of the eyepieces. Then you can at least see where you are... I will do this exemplarily for my telescopes on another page.
Examples
The beneficial magnification is defined as the magnification above which further magnification does not provide any new information. This is also referred to as "empty" magnification. Nevertheless, according to Stoyan, it can make sense to go beyond the beneficial magnification to the maximum magnification. This applies in particular to structures (e.g. on the moon) for which the standard resolution formulas (Dawes, Rayleigh) do not apply. According to my experience, some structures only become recognizable when they have reached a certain size, no matter whether this is done with empty magnification or not...
The use of high magnifications of course depends very much on the quality of the sky (seeing), and often it is not possible to use high magnifications. As a rule of thumb, I have read that on most days in Central Europe, a magnification beyond 200 x does not make any sense. With my telescopes, the limit often seems to be reached already at 100 x...
A starfriend wrote me that particularly in the range of high magnifications, it is important to have several finer graduated eyepiece focal lengths available. Depending on the seeing, on certain days, a slightly longer focal length eyepiece (e.g. 12 mm) can mark the "boundary", whereas on other days it may also be possible to use shorter focal lengths (e.g. 8 mm or even 4 mm). He often switches between the eyepieces, until he has found the best eyepiece for the day. In his opinion, having only three eyepieces for one telescope is not enough: He needs one for the overview, a medium magnification, and several ones for fine tuning depending on seeing for the higher magnifications.
In practice, it is, of course, important to know, which magnification fits which purpose. I found (and added to... source regrettably unknown...) the following recommendations for the use of magnifications:
In principle, these magnification values might be used to select the appropriate focal lengths of eyepieces for a particular telescope according to the following formula (transformation of the above formula):
Apparently, however, it is more common to use the exit pupil as a criterion for selecting focal lengths of eyepieces, because the brightness of the image is also taken into account. For example, a starfriend told me that he does not like the approach of using just the magnification that much. One should always consider the exit pupil and be aware that besides the appropriate magnification, also the brightness of the image has a decisive part in the observational success.
In progress
The apparent field of view (viewing angle) determines the angle that is shown by an eyepiece as a section of the sky. It depends on the type of the eyepiece an is usually given by the manufacturer of the eyepiece. Here are some examples:
Eyepieces with a field of view up to 55° are often characterized as providing "tunnel vision." Eyepieces with a field of view of 80° and more are often advertised as that you are "floating in front of objects in space." Eyepieces with a viewing angle around 70° are considered as "ideal for the human eye" and as "optimal for observation"  with larger fields of view you have to "look around the corner" to overlook the whole field of view. More about eyepiece types can be found on Wikipedia and on page Some Information About Eyepieces.
The true field of view (angle of view/vision, visual angle) determines the size of an object as it can be observed in a telescope:
It is thus, obtained by dividing the apparent viewing angle by the magnification (e.g., 70 degrees / 100 = 0.7 degrees).
Why is the true visual angle important? The size of objects in the sky is indicated in angular degrees, that is, as viewing angles (for example, the sun and the moon both have a visual angle of 0.5 ° or 30'). Such data can be found, for example, in books about deep sky objects. Two unfavorable situations may arise when observing sky objects:
When choosing eyepieces, it is therefore good to know how the visual angle of sky objects and the field of view (the true visual angle) of an eyepiece relate to each other.
Older Kellner and orthoscopic eyepieces have an apparent viewing angle of 40 degrees, Plössl eyepieces one of 50 degrees or a little more, and above them, the wideangle, super and ultrawideangle eyepieces start, which have become increasingly popular in recent years. According to the above criteria, there is no difference between the different types of eyepieces: all have the same magnification and exit pupil. In practice, however, according to advertising, this means a difference between "tunnel vision" and "space walk"  which is where the angle of view of the eyepiece comes into play... To put it simply, the larger the apparent viewing angle, the larger the true viewing angle and thus, the section of the sky that can be observed in the eyepiece.
By the way, eyepieces with an apparent viewing angle of 70 degrees are regarded as "optimal" for our eyes, because at this viewing angle you do not have to wander around with your eyes and you can see everything at a glance.
The (true) angle of view is a measure of the size of the objects with which they can be observed in the telescope. It is obtained by dividing the apparent viewing angle by the magnification:
Example: 70 degrees / 100 = 0.7 degrees
Why is knowledge of the true angle of view useful? The size of celestial objects is given in degrees, that is, as the angle of view (for example, the sun and the moon have an angle of view of 0.5° and 30', respectively). Such measures can be found e.g. in books about deep sky objects.
This can result in two unfavourable situations:
When selecting an eyepiece, it is therefore good to know how the angle of view of the celestial object and the field of view (the true angle of view) of the eyepiece relate to each other. Moreover, if you have to decide between a "normal" eyepiece and a (U)WA eyepiece, you are faced with the following alternatives:
When choosing a (U)WA eyepiece, you have to decide how large the viewing angle should be. I already find a viewing angle of 60 degrees quite pleasant, 70 degrees even more so, and with my 80 degrees eyepieces, I have to "wander around" a bit with my eyes, which I do not find pleasant at all... In addition, eyepieces with a viewing angle of 100 to 110 degrees are heavy and very expensive. I feel that the 80s are still "just affordable"... Since I have never looked through a 100110 degree eyepiece, I cannot tell you anything about the "space walk" experience. But there are starfriends who do not want to miss this experience anymore...
23.04.2019 