Introduction and Rehearsal...  Exit Pupil as A Criterion for the Selection of Focal Lengths for Eyepieces  Applied to My Eypieces...  Discussion of the Results  References
On this page, I discuss "theoretical" criteria for the choice of the focal length of eyepieces, such as the exit pupil, magnification, and visual angle. Obviously, mostly the exit pupil criterion is been used for this purpose. Here, I present two variants that are based on the exit pupil and apply them to my own telescopes. Visitors should have no trouble applying this to their own equipment.
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 two recommendations, would like to introduce these and apply them to my telescopes and eyepieces further down. Magnification and visual angle (field of view) are also involved in all this, but these do not seem to be used as a primary criterion (as far as I can tell so far ...).
Because the terms magnification, exit pupil, and visual angle play an important role on this page, I would like to start my discussion with repeating a few things that I already wrote elsewhere on this Website.
Please note that, unfortunately, the problem gets somewhat harder, if one possesses several telescopes...
The magnification of a telescope can be calculated as follows:
For a given telescope, it depends only on the focal length of the eyepiece in use and can be easily calculated in your head. In practice, it is, of course, important to know, which magnification fits which purpose:
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):
Obviously, however, it seems to be more common to use the exit pupil.
The definition of the exit pupil is somewhat complex and can be found in the glossary.
General rules for the exit pupil are:
I found two recommendations that suggest, which exit pupil is suitable for which (deep sky) objects / uses. The magnification and the focal length of the eyepiece can then be calculated from this. I summarized these proposals in a table (adapted):
Ronald Stoyan  penumbra  
Deep Sky Application Area  Exit Pupil  Application Area  Exit Pupil  
Search of objects, largearea nebulae  7 
Maximum overview for largearea nebulae  4.5  6 

Nebulae, star clusters  4 
Galaxies and midsize deep sky objects  2  3 

Galaxies, globular clusters  1.5 
Maximum magnification for moon and planets  0.6  1 

Planetary nebulae, small galaxies  0.7 

Double stars, small planetary nebulae  0.4 
In the left table, there are criteria for the eyepiece choice that Ronald Stoyan provides in his book Deep SkyReiseführer (travel guide). There, he does, however, not mention explicitly that these numbers represent the exit pupil (actually, he provides formulae to calculate the magnification, see further down). In a discussion thread, I found similar, but slightly simpler suggestions for the eyepiece selection from poster penumbra. He specifies only three applications (corresponding to 3 eyepieces), but he provides values ranges for the exit pupil (see the table on the right).
The application of these recommendations is described below.
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.
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 measures can be found, for example, in books about deep sky objects. Two unfavorable situations may arise:
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.
In the following, I will only consider the case of the exit pupil as a selection criterion for the focal length of eyepieces. In his book Deep Sky Reiseführer, Ronald Stoyan provides the criteria for the selection of the focal lengths of eyepieces presented in the table below. He does, however, not mention explicitly that the numbers in the denominators correspond to the exit pupil (I mentioned already similar criteria provided by penumbra and will apply them below).
Deep Sky Application Area  Exit Pupil  Calculate Magnification from 
Search of objects, largearea nebulae  7 
Aperture / 7 
Nebulae, star clusters  4 
Aperture / 4 
Galaxies, globular clusters  1.5 
Aperture / 1.5 
Planetary nebulae, small galaxies  0.7 
Aperture / 0.7 
Double stars, small planetary nebulae  0.4 
Aperture / 0.4 
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 in the range between minimum and maximum usable magnification). Stoyan emphasizes the importance of extending the range of magnifications to the minimum and maximum magnifications. However, according to Stoyan, with small opening ratios (f15  f8), the minimum magnifications cannot be realized because there are hardly any eyepieces with focal lengths of more than 40 mm available.
*) Or simply "aperture" as Stoyan puts it
In the following, I apply the criteria from Stoyan and penumbra to my telescopes (as well as to a 150/900 telescope with f6, that is, the same focal ratio as my GSD 680 has). For a better overview, I compile the results in tabular format.
In the following table, I calculate the focal lengths and magnifications in addition to the exit pupil of 7 also for an exit pupil of 6.5.
Criteria by Stoyan  Exit Pupil 
Focal Length
of Eyepiece 
Exit Pupil 
Magnification 

Deep Sky Application  GSD 680  150/900  P130  100P  102  GSD 680  150/900  P130  100P  102  
f > 
6 
6 
5 
4 
12.75 
Focal Length > 
1200 
900 
650 
400 
1300 

Search for objects, largearea nebulae  7 
42.0 
42.0 
35.0 
28.0 
89.3 
7 
28.6 
21.4 
18.6 
14.3 
14.6 

Ditto, exit pupil for older people  6.5 
39.0 
39.0 
32.5 
26.0 
82.9 
6.5 
30.8 
23.1 
20.0 
15.4 
15.7 

Nebulae, star clusters  4 
24.0 
24.0 
20.0 
16.0 
51.0 
4 
50.0 
37.5 
32.5 
25.0 
25.5 

Galaxies, globular clusters  1.5 
9.0 
9.0 
7.5 
6.0 
19.1 
1.5 
133.3 
100.0 
86.7 
66.7 
68.0 

Planet ay nebulae, small galaxies  0.7 
4.2 
4.2 
3.5 
2.8 
8.9 
0.7 
285.7 
214.3 
185.7 
142.9 
145.7 

Double stars, small planetary nebulae  0.4 
2.4 
2.4 
2.0 
1.6 
5.1 
0.4 
500.0 
375.0 
325.0 
250.0 
255.0 

Check  F/EP 
Focal
Length of Eyepiece 
EP/F 
Magnification 

Minimum (Factor)  1.5 
4.0 
4.0 
3.3 
2.7 
 
Minimum (EP) 
6.5 
30.8 
23.1 
20.0 
15.4 
15.7 

2 
3.0 
3.0 
2.5 
2.0 
6.4 
7 
28.6 
21.4 
18.6 
14.3 
14.6 

Maximum (EP)  6.5 
39.0 
39.0 
32.5 
26.0 
82.9 
Maximum (Factor) 
1.5 
300 
225 
195 
150 
 

7 
42.0 
42.0 
35.0 
28.0 
89.3 
2 
400 
300 
260 
200 
204 

3* 
600 
450 
390 
300 
300 
*) According to Stoyan maximum for small extended objects (according to Stoyan, the maximum magnification is: Aperture *2/0.7)
As can be seen from the table above, these rules exploit the minimum magnifications on one end, whereas they go beyond the maximum beneficial magnification on the other end. In his book, Stoyan explains why, under certain conditions (e.g., small, flat objects), it may be useful to exceed the maximum beneficial magnification. According to him, this can be the twice the beneficial magnification (aperture / 0.7, which is somewhat below the most frequently used factor of 1.5). A roughly simplified or somewhat optimistic approach might therefore use a factor of 3. Stoyan mentions, however, that the maximum magnification cannot be achieved for certain telescope types, such as Newton and MaksutovCassegrain telescopes, because of the inevitable obstruction (given by the secondary mirror and its hold). Anyway,with this information, the magnification values for an exit pupil of 0.4 mm appear to be achievable (they are clearly below the factor of 3).
I comment on the results of this table with regard to my eyepieces below.
These specifications by penumbra lead to the following table with the values of my telescopes (reverse order for easier comparison against the original):
Criteria by penumbra  Exit Pupil 
Focal
Length of Eyepiece 
Exit Pupil 
Magnification 

GSD 680  150/900  P130  100P  102  GSD 680  150/900  P130  100P  102  
Application Area  f > 
6 
6 
5 
4 
12.75 
Focal Length> 
1200 
900 
650 
400 
1300 

Maximum overview for largearea nebulae 
from  6 
36 
36 
30 
24 
76.5 
from  6 
33.3 
25.0 
21.7 
16.7 
17.0 

to  4.5 
27 
27 
22.5 
18 
57.4 
to  4.5 
44.4 
33.3 
28.9 
22.2 
22.7 

Galaxies and midsize deep sky objects 
from  3 
18 
18 
15 
12 
38.3 
from  3 
66.7 
50.0 
43.3 
33.3 
34.0 

to  2 
12 
12 
10 
8 
25.5 
to  2 
100.0 
75.0 
65.0 
50.0 
51.0 

Maximum magnification for moon and planets 
from  1 
6 
6 
5 
4 
12.8 
from  1 
200.0 
150.0 
130.0 
100.0 
102.0 

to  0.6 
3.6 
3.6 
3 
2.4 
7.7 
to  0.6 
333.3 
250.0 
216.7 
166.7 
169.9 

Check  F/EP 
Focal
Length of Eyepiece 
EP/F 
Magnification 

Minimum (Factor)  1.5 
4.0 
4.0 
3.3 
2.7 
 
Minimum (EP) 
6.5 
30.8 
23.1 
20.0 
15.4 
15.7 

2 
3.0 
3.0 
2.5 
2.0 
6.4 
7 
28.6 
21.4 
18.6 
14.3 
14.6 

Maximum (EP)  6.5 
39.0 
39.0 
32.5 
26.0 
82.9 
Maximum (Factor) 
1.5 
300 
225 
195 
150 
 

7 
42.0 
42.0 
35.0 
28.0 
89.3 
2 
400 
300 
260 
200 
204 

3 
600 
450 
390 
300 
300 
*) According to Stoyan maximum for small extended objects (according to Stoyan, the maximum magnification is: Aperture *2/0.7)
In the following, I comment on the results of this and the first table with regard to my eyepieces.
Here is an overview of my eyepieces that are suitable for a choice, although I would like to use only the eyepieces in the top row:
As always, I created a table to make the results  hopefully  easier to understand:
Telescope  Eyepieces according to Stoyan  Useful Choice?  What Is Missing?  What Do I Like to Use? 
GSD 680**  2.4* / 4.2 / 9 / 24 / 3942  4 / 9 / 25 / 3032  40, 25 of better quality?  6?, 7 
P130  2.0* / 3.5 / 7.5 / 20 / 32.535  4 / 7 / 16?20 / 32  2*?, 20 of better quality?  6? 
100P  1.6* / 2.8 / 6.9 / 16 / 2628  4 / 7 / 16 / 2532  3?  6? 
102  5.1* / 8.9 / 19.1 / 51 / 82.989.3  6*7 / 9 / 1620 / 32  5*?, 20 of better quality?, 40?  
Telescope  Eyepieces according to penumbra  Useful Choice?  What Is Missing?  What Do I Like to Use? 
GSD 680**  3.66 / 1218 / 2736  4, 16, 3032    6?, 7 
P130  35 / 1015 / 22.530  4, 16?, 32?  1012 of better quality?  7 
100P  2.44 / 812 / 1824  4, 910, 16?2025  910 und 2025of better quality?  7 
102  7.712.8 / 25.538.3 / 57.476.5  910, 2532  910 of better quality?, 40?  6**, 7, 16 
*) Beyond beneficial magnification; **) also valid for a 150/900 telescope; italic: eyepieces of lower quality
Looking at the focal length suggestions on the basis of the criteria by Stoyan and penumbra, I feel that they do fit together well. My impression is that the requirements by penumbra fit somewhat better my existing eyepieces than those by Stoyan (except perhaps the Heritage 100P). According to penumbra, my 7 mm eyepiece seems already to be not needed, but I like to use it; it also achieves almost the maximum magnification (factor 2) with the Skymax102 OTA  and for that reason, I bought it.
If I would recognize gaps in my eyepiece set, then this would be in the ranges of 912 mm and 2025 mm. Here, I own eyepieces, but their quality is rather low.
According to Stoyan's creiteria, there are gaps at the short end, but I have doubts that the seeing at my place of residence is sufficient for such large magnifiacation (the same might be valid for the quality of my telescopes ...). I also regard the gap at the long end with skepticism. I do not know if I should close this gap, because with my former 40erPlössl I had problems with the insight due to my age.