Magnification

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Telescope Magnification | Influencing Variables, Properties, and Limits | Types of Magnification | "Detour" via the Dawes Criterion to the "Common Denominator"... | Bottom Line | Links

On this page I try to make the topic "magnification of telescopes" more understandable for me and to clarify what, besides the "actual" magnification, other magnification terms used in this context do actually mean.

 

Telescope Magnification

The beginning is simple and can be read in many textbooks and other sources. The following definition is based on Gehrtsen (physics textbook):

It seems to be the ratio of the tangens values of these angles, if you want to be exact, but for small angles the angle in radians is a good approximation for the tangens.

This relation has to be expressed in "telescope parameters" in order to be usable in practice. For objects that are infinitely far away, the magnification of a telescope is calculated from the focal length of the telescope and the focal length of the eyepiece (see e.g. Gehrtsen or here; in German):

A telescope therefore does not have a "magnification per se", but this results from the interaction of telescope and eyepiece and is therefore not "fixed".

Furthermore applies (without derivation):

According to formula 1 for the magnification, one might make the magnification as high (and as low) as one likes. However, there are a number of properties or limits that put an end to this:

Below I will discuss some of these influencing variables/limits in more detail.

Further Magnification Terms

There are also a number of other magnification terms for telescopes that are "fixed" because they depend essentially on the aperture of the telescope (or, depending on your point of view, on its resolution, which in turn depends on the aperture). I would like to introduce the following terms:

Certainly there are other terms in this context. On the Internet and in the literature I have found that the presented terms are used by different authors in different ways, or with overlapping meanings, or that they lead to different values although the definitions of the terms are the same. In the following, I try to bring some "light into the darkness" for myself...

Why were these additional magnification terms introduced? I suppose they were introduced to give users some orientation on the scale from no magnification at all to infinitely high magnification (although the different uses of the terms seem to be rather confusing...). They refer to certain properties and limits of telescopes and the eye; in addition there are limits, such as air turbulence, which are independent of all this. Therefore, before describing the magnification terms, first a few remarks about influencing variables, properties and limits!

 

Influencing Variables, Properties, and Limits

Controlling the Amount of Light: Entrance Pupil/Aperture and Exit Pupil

The amount of light falling on the eye is determined by the "openings" of the telescope on the objective and eyepiece ends:

The aperture is a characteristic of a telescope that is determined by its construction, and perhaps it is its most important property of all, because it determines some other telescope parameters, such as light gathering power and resolution, exclusively, as well as other parameters (aperture ratio, magnification, ...) in conjunction with other parameters (focal length, eyepiece focal length, ...).

The exit pupil of a telescope or eyepiece thus determines the maximum amount of light that can reach the eye:

In the "eyepiece language" this is means:

The exit pupil can be determined by the magnification and the aperture of the telescope, but also by the focal length of the eyepiece and the aperture ratio of the telescope, so it is not an "eyepiece property":

This formula (formula 3) can be transformed, for example, in order to calculate the (beneficial) magnification using the exit pupil (see below).

For a given telescope the entrance pupil/aperture of which is fixed and predetermined (by the primary mirror or the objective lens), the exit pupil of the eyepiece controls the amount of light reaching the eye, at least as long as it is equal to or smaller than the eye pupil. This is done in practice by using different eyepieces and thus different magnifications. The magnification types still to be discussed tell you, among other things, within which range you should use magnifications. However, they do not tell you which magnification or exit pupil is most suitable for which celestial object; this has to be be taken from other sources. They also do not tell you which maximum magnification the current state of the sky (the seeing) allows.

By the way: According to formula 3, telescopes with larger apertures have the advantage over those with smaller apertures that their exit pupil is larger at the same magnification, namely by the factor by which the apertures differ. This allows them to present planar objects brighter as long as these are not larger than the field of view (according to Stoyan, Deep Sky Reiseführer). Here an example calculation with two telescopes having aperture_1 and aperture_2:

In other (or my...) words: The exit pupils behave like the apertures.

Resolution of the Human Eye

The resolution of the human eye is another important parameter or limit in the following considerations. So here are a few data that I have collected (according to Stoyan, Deep Sky Reiseführer, and Spektrum der Wissenschaft):

Resolution of Telescopes

The resolution capability (or only resolution) of a telescope is defined as the ability to display two closely spaced objects (e.g. double stars) separately. Two criteria have been developed for this purpose:

In practice, "rules of thumb" are used for both criteria (that is why I leave out the complex mathematics behind):

The resolution depends solely on the telescope aperture (objective lens or mirror diameter). Obviously, the Dawes criterion is more "practical" than Rayleigh's (see the article by Stefan Gotthold, in German), and even the manufacturers usually only provide this value (probably because it looks "better").

Why do I list the resolution here? On the one hand, as we will see further below, at the in the beneficial magnification, the resolution of the telescope matches that of the eye. On the other hand, other types of magnification are also associated with the resolution because they depend on the aperture.

Sky Quality, Seeing

The sky quality depends on the degree of light pollution and is measured on a scale of 9 steps. It essentially determines the limiting stellar magnitude. Therefore it is less relevant in the context of magnification types.

The so-called seeing (air turbulence, air movement) can disturb the image in the telescope to such an extent that an observation above a certain magnification makes little or no sense. In the literature and on the Internet you will find simple rules for maximum magnification under different conditions based on practical experiences. I have read the different recommendations for upper magnification limits and list some here without any comments:

In other words: No matter which maximum magnification the formulas below provide, if the quality of the sky is low, the seeing of the sets an upper limit to the maximum useful magnification. This upper limit also depends on the object being observed (i.e. whether it is a double star, a two-dimensional object, a planet or the moon).

By the way, the seeing depends on the square of the aperture, because it is a function of the area (Stoyan, Deep Sky Reiseführer). This is why small telescopes reach their maximum magnification more often than large ones.

Limits Due to the Construction

Telescopes cannot be built to any size for technical reasons, also because the size of the lenses and/or mirrors reaches technological and weight limits. But the hobby astronomer himself sets limits here as well, especially with regard to the weight. For me, because of the weight, an aperture of 6" (150 mm) is already the upper limit (before that I had tried 8" and 10"...). In the hobby area, the apertures cover a range from 60 mm to a more than 500 mm (20"), which roughly corresponds to a ratio of 1:10.

For technical reasons, eyepieces can also only cover a certain focal length range. In the hobby area, this is between 2 mm and 40 mm (please understand this only as an orientation!). Thus, a range of approximately 1:20 is covered! If you add focal length reducers and extenders, you can increase this range even further, for example by a factor of 15...

In fact, not every telescope aperture can be combined with every eyepiece focal length. I have no idea, how large the actual magnification range is in the hobby area...

 

Types of Magnification

In the following, I present a number of magnification terms or types that I have found in literature, on the Internet, and in advertisements.

Normal Magnification

I found the term normal magnification in three meanings:

Minimum Magnification, Minimum Usable Magnification, Normal Magnification

The minimum magnification is defined by the fact that the diameter of the telescope's exit pupil is the same as that of the eye pupil (our "entrance pupil"). It follows from this that it depicts planar objects with maximum brightness, because as soon as the exit pupil of the telescope is smaller than the eye pupil, light is lost and planar objects are attenuated (according to Stoyan, Deep Sky Reiseführer).

Values between 5 and 8 mm are given for the size of the eye pupil. The smaller values are for older people, but I have also found a source that gives 7 mm for 70-year-olds... Typically, the minimum magnification is calculated for an exit pupil/eye pupil of 6.5 or 7 mm.

As far as the telescope is concerned, the minimum magnification depends only on its aperture (and indirectly on its resolution). The minimum magnification corresponds to the maximum (reasonable) usable focal length of the eyepiece, which can be determined using the magnification formula (Formula 1) after inserting the minimum magnification in order to select your eyepieces accordingly (if such long focal length eyepieces exist at all...):

Beneficial Magnification

The beneficial magnification of a telescope is defined by the fact that the resolution of the telescope and that of the eye match exactly*. It is the highest magnification at which there is still a gain in detail**. Only at this magnification, the resolution and light gathering capacity of the telescope are fully exploited and its limiting visual stellar magnitude reached. Beyond that no further details become visible and one speaks of "empty magnification".

) This can be interpreted as follows: the telescope resolution is magnified exactly to the extent that it has the same value as the resolution of the eye.
**) The details result from the assumed resolution of the eye, more about this later.

As far as the pure definition is concerned, most sources still agree, but then the differences begin. Different sources assume, for example, different values for the resolution of the eye, mostly without specifying this explicitly: the majority of sources assume a resolution of 2' (Astroshop.de, for example, under the label "normal magnification", without specifying the resolution), but some assume a resolution of 3' (Stoyan, Deep Sky Reiseführer, ...). This leads to a difference in the magnification of at least 50%, as the following formulas exemplify:

With an assumed resolution of the eye of 2', the exit pupil is exactly 1 mm (which is sometimes indicated), with one of 3' it is only 0.67 mm. Stoyan (Deep Sky Reiseführer) assumes a resolution of about 3' and provides a value of 0.7 mm for the exit pupil. This results in a magnification according to formula 2:

This value is close to the value determined with the factor 1.5, but unfortunately not exactly (1/0.7 = 1.43; 1/0.67 = 1.5; however, Stoyan does not specify 0.67 as the exact value for the exit pupil, he uses 0.645 mm: 1/0.645 = 1.55)... Well, one should not take such "rough calculations" too seriously anyway!

If one chooses a value of 3' or even 4' for the resolution of the eye, one however gets already into a magnification range, which is also called maximum usable/sensible (or: maximum usable) magnification. I deal with this in the following!

Maximum Usable Magnification

The maximum usable magnification of a telescope is often also defined by the fact that the resolution of the telescope and of the eye are exact match each other. It is also characterized as being the highest magnification at which there is still a gain in details (beyond this, no further details become visible and one speaks of an "empty magnification").

The maximum usable magnification depends essentially on the aperture (diameter of the primary mirror or lens opening) of the telescope:

The factor X amounts to:

For X = 2, the exit pupil amounts to exactly 0.5 mm at this magnification, for X = 1.5 it amounts to 0.67 mm (see above).

According to the explanations for the beneficial magnification, these are, in my opinion, actually only variants of the beneficial magnification, namely those with a lower assumed resolution of the eye, that is, 3' and 4' (see also below). In other words, for other authors, Stoyan's beneficial magnification is, at least for some telescope types, already the maximum usable magnification. For other telescope types, they apply a factor of 2. Sometimes, a factor of 2 is also given for all telescope types, which look better when advertising telescopes... Stoyan, on the other hand, assumes the lower value of almost 1.5 for all telescope types. In the end, one might say that, for Stoyan, the exit pupil is the "measure of all things." This can also be concluded from his eyepiece recommendations in the Deep Sky Reiseführer.

Maximum Magnification

As just described, a factor of X = 2 is often given for the maximum usable magnification. In his article "Vergrößerung und Grenzgröße im Teleskop – Fallbeispiel M 13 (Magnification and limiting magnification in telescopes - case study M 13)" on the astronmie.de Website Stoyan writes: "There is no optically defined maximum magnification for flat objects - rules like "double the objective diameter" (see above) are without fundament." This makes me wonder whether the maximum usable magnification can be "discarded," particularly since Stoyan's "accompanying" factor X = 1.5 was already assigned to the beneficial magnification. If I understand him correctly, it looks much like this...

Accordingly, Stoyan takes a more "generous" approach to the topic of "maximum magnification." According to his Deep Sky Reiseführer, it is often better for secure perception to exceed the beneficial magnification, for example, in order to increase the distance between stars. Even with smaller flat objects, it makes sense for Stoyan to go far beyond this. According to him, the so-called maximum magnifications are therefore 2 times higher than the beneficial magnifications. Simplifying I therefore use the factor X = 3 for the maximum magnification:

At maximum magnification, the exit pupil amounts, according to Stoyan, to 0.35 mm, being already fairly small, and the resolution of the eye to 6'.

The seeing sets the limit for the maximum magnification, particularly for telescopes with large apertures. For examples, see above.

Intermediate Status...

A definition, various types of magnification that use it and partly overlap in their meaning, partly lead to different results, while different types of magnification lead to the same result, in addition to different resolution of the human eye - how can all this be put together? This is exactly what I would like to attempt in the next section!

 

"Detour" via the Dawes Criterion to the "Common Denominator"...

In the above remarks, the resolution of the human eye is mentioned again and again, so that I have come to the conclusion that this is the central variable in describing the different types of magnification (the exit pupil plays a similar role, but I regard it as a "derived" variable). Therefore, I have searched for a "common denominator" for all types of magnification and found it (as I believe...) by making a detour via the Dawes formula for the resolution of a telescope. This approach makes the resolution of the human eye explicit and thus, the central variable.

I start from the beneficial magnification, which is characterized by the fact that the resolution of the telescope is matched to that of the eye. This means that the resolution of the telescope has to be multiplied by a magnification factor (in the following simply called "magnification"), so that the resolution of the human eye results:

If one resolves the equation for the beneficial magnification, this results in:

The Dawes formula for the resolution of the telescope (the most commonly used form of resolution) is:

The Dawes formula used in the formula for the beneficial magnification yields:

Often the Dawes formula is simplified too:

(I have also seen values of 115 or 117...) With this, for easier calculating with minutes and seconds, we get approximately:

Resolution of the Eye and Types of Magnification

This leads me "formally" back to the known relationship for the beneficial magnification, because when I use 120" (2') for the resolution of the eye, exactly the relationship given above for 2' results: the beneficial magnification corresponds numerically to the aperture of the telescope in mm. I can also use it to describe the different maximum magnifications if I insert the corresponding assumed resolution of the eye into the formula. In this way, I make the assumed resolution of the human eye explicit in the formula, which rarely happens in practice. In other words, the various magnifications listed above can all be described by a specific value for the resolution of the human eye:

I suppose that I only "rediscovered" what everyone already knows, but many sources do not explicitly write this down. And that is why I had to find out myself...

What else is to be done now? In my opinion, it is now a matter of understanding more precisely what the different values for the resolution of the human eye mean in practice. For this, I have looked around again in the literature and found the following information about "sky structures." The basic definition, together with the determined magnification values, then applies, so to speak, only to the corresponding structures. Here is a list of what I have found so far (in progress):

In other words, when you use strained vision, you need less magnification than when you use comfortable vision, when you want to separate double stars, less than when you want to recognize bright areas or even smaller bright areas, etc. and so on.

Obviously, the standard definition of maximum usable magnification refers to the separation of double stars (2'), whereas Stoyan's beneficial magnification refers to bright areas (and his maximum magnification to small bright areas). This is exactly what Stoyan writes in his Deep Sky Reiseführer.

A further todo would be to clarify which consequences result from the different exit pupils, which correspond to the different values of the resolving power...

 

Bottom Line

And how Do I Cope with this "Chaos"?

For the normal magnification, as well as for the beneficial magnification and the maximum usable magnification (and its variants) I have read that this is the magnification which shows maximum details and that, beyond that magnification, you only get "empty" magnification (which can also be useful because it magnifies small details and thus, sometimes makes them visible at all). This has confused me to a certain extent!

A first hint to an approach that unites everything was given by the fact that some sources indicate the underlying resolution of the eye and/or the exit pupil, sometimes explicitly, sometimes hidden or indirectly. The resolution of the eye varies between 1' and 3', the exit pupil between 2 mm and 0.67 mm, accordingly. Obviously, the above statement "maximum details" or the definition "equal resolution of eye and telescope" is to be understood on the basis of the resolution of the human eye which the respective source uses. Here once again as a reminder:

The magnifications presented here beyond the minimum magnification thus represent a continuum of increasing assumed resolution of the human eye (from 1' to 6'), which makes certain structures in celestial objects only visible at all. On the other hand, the exit pupil decreases from 2 mm (or from 6-7 mm, if the minimum magnification is included) to 1/3 mm, where you can hardly see anything anyway, if I understand this correctly... The minimum magnification is the magnification at which the exit pupil is at maximum (6-7 mm), that is, equal to the eye pupil (and the resolution seems to be close to the theoretical limit...).

Televue Website on the Exit Pupil (adapted and translated)

"Normal Magnification"

"Maximum Magnification"

My Bottom Line...

Basically, it does not matter what different authors mean by the individual magnification types, as long as one knows on which assumptions they base their statements. In any case, one can also consider the whole matter from the point of view that the numerical value of the aperture (in mm) represents a "good standard/usage magnification", 1.5 * aperture (in mm) represents the "maximum standard/usage magnification" and 3 * aperture (in mm) represents the "absolute maximum that can be achieved under certain circumstances". The brightest image in the eyepiece is offered by the minimum magnification.

And a magnification that corresponds to twice the numerical value of the aperture (in mm) (resolution 4', exit pupil 0.5 mm) should perhaps be perceived as a little "too ambitious for "standard use" and remember that, according to Stoyan, there is no evidence to regard this particular magnification as maximum magnification...

However, as before, I will operate my telescopes by default with maximum magnifications of 1.5 to 2 times the numerical value of the aperture (in mm) and try larger factors on the moon, and perhaps also on planets (up to 3 to 4 times the numerical value of the aperture in mm).

 

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12.04.2019